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Essays on Forecasting with Linear
State-Space Systems
2016-2
Lorenzo Boldrini
PhD Thesis
DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS
AARHUS UNIVERSITY DENMARK
ESSAYS ON FORECASTING WITH LINEAR
STATE-SPACE SYSTEMS
By Lorenzo Boldrini
A PhD thesis submitted to
School of Business and Social Sciences, Aarhus University,
in partial fulfilment of the PhD degree
in Economics and Business Economics
February 2016
CREATESCenter for Research in Econometric Analysis of Time Series
PREFACE
This dissertation is the result of my PhD studies at the Department of Economics
and Business Economics at Aarhus University and was written in the period from
September 2012 to August 2015. I am grateful to the Department of Economics and
Business Economics as well as CREATES - Center for Research in Econometric Analy-
sis of Time Series-DNRF78 -, funded by the Danish National Research Foundation,
for providing both a unique research environment and financial support.
Several people deserve my gratitude. In the first place, I wish to thank my main
advisor, Prof. Eric T. Hillebrand for the support, advice and encouragement he has
given me during these three years. Working with Eric has never failed to be challeng-
ing and thought provoking. Thank you to my co-advisor and Center Director, Prof.
Niels Haldrup and the Center Administrator Solveig Nygaard Sørensen, for creating
a great research environment and for the many interesting PhD courses that were
organized by CREATES during my studies. I also wish to thank Prof. Eduardo Rossi
and Prof. Maria E. De Giuli for encouraging me to apply for a PhD position.
From January 2015 to April 2015 I had the great pleasure of visiting Prof. Siem
Jan Koopman at the Department of Econometrics and Operations Research at VU
Amsterdam. I had the opportunity to have frequent, helpful meetings with Siem Jan,
who never lacked enthusiasm for the subject at hand and always offered a diverse
perspective on the topic.
At Aarhus University I would like to thank the faculty and staff at the Depart-
ment of Economics and Business Economics. In particular, I would like to thank
Ulrich, Johannes, Nima, Matt, Ulises, Anders Kock and Henning B. for their help
on some computational and analytical aspects involved in this work and Solveig,
Mikkel, Jonas and Niels S. for their editorial help involved in the writing of this thesis.
I am very grateful to my fellow PhD students for the many interesting conversations
and enjoyable activities we shared and in particular to Camilla, Silvia, Alice, Andrea,
Silvana, Vladimir and Eduardo. Special thanks go to Federico, Orimar and Hassan. I
am indebted to them for the copious fruitful discussions we had and fun moments
we shared. Lastly, I am thankful for the support my family and friends have shown
me during the last three years.
Lorenzo Boldrini
Aarhus, August 2015
i
UPDATED PREFACE
The pre-defence meeting was held on the 25th November 2015, in Aarhus. I am grate-
ful to the members of the assessment committee consisting of Siem Jan Koopman
- VU University Amsterdam -, Tommaso Proietti - University of Rome Tor Vergata
- and Asger Lunde - Aarhus University and CREATES - for their careful reading of
the dissertation and their many insightful comments and suggestions. Some of the
suggestions have been incorporated into the present version of the dissertation while
others remain for future research.
Lorenzo Boldrini
Aarhus, February 2016
iii
CONTENTS
Summary vii
Danish summary xi
1 Supervision in Factor Models Using a Large Number of Predictors 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Forecasting with dynamic factor models . . . . . . . . . . . . . . . . . 5
1.3 Quantifying supervision . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 The Forecasting Power of the Yield Curve 41
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Dynamic factor models and supervision . . . . . . . . . . . . . . . . . 45
2.3 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3 Forecasting the Global Mean Sea Level 75
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
v
vi CONTENTS
3.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
SUMMARY
This dissertation is composed of three self-contained, independent chapters. The
statistical framework in all three chapters is that of linear, Gaussian, state-space
systems. In the first two chapters, we study a method to forecast macroeconomic
time series whereas the third one is focused on the forecasting of the global mean sea
level, conditional on the global mean temperature.
More in detail, in Chapter 1 we investigate the forecasting performance of a partic-
ular factor model in which the factors are extracted from a large number of predictors.
We use a semi-parametric state-space representation of the factor model in which the
forecast objective, as well as the factors, is included in the state vector. The factors are
informed of the forecast target (supervised) through the state equation dynamics. We
propose a way to assess the contribution of the forecast objective on the extracted fac-
tors that exploits the Kalman filter recursions. We forecast one target at a time based
on the filtered states and estimated parameters of the state-space system. We assess
the out-of-sample forecast performance of the proposed method in a simulation
study and in an empirical application to US macroeconomic data. We compare our
specification to other multivariate and univariate approaches and use the Giacomini
and White (2006) test to assess the relative conditional predictive ability of the mod-
els. In particular, we choose as competing models: principal components regression,
partial least squares regression, ARMA(p,q) processes, an unsupervised factor model,
and a standard dynamic factor model with separate forecast and state equations.
We find that, variables which contribute more to the variance of the filtered states,
are the ones which benefit more from the supervised framework and vice versa. The
proposed specification performs particularly well in forecasting the federal funds
rate, the unemployment rate, and real disposable income.
In Chapter 2, we study the forecast power of the yield curve for macroeconomic
time series. We employ a state-space system in which the forecasting objective is
included in the state vector. This amounts to an augmented dynamic factor model in
which the factors (level, slope, and curvature of the yield curve) are supervised for
the forecast target. In other words, the factors are informed about the dynamics of
the forecast objective. The factor loadings have the Nelson and Siegel (1987) struc-
ture and we consider one forecast target at a time. We forecast the consumer price
index, personal consumption expenditures, the producer price index, real disposable
vii
viii CONTENTS
income, unemployment rate, and industrial production. All time series are relative to
the US economy. We compare the forecasting performance of our specification to
benchmark models such as principal components regression, partial least squares
regression, ARMA(p,q) processes, and two-step forecasting procedures based on
factor models. We use the yield curve data from Gürkaynak, Sack, and Wright (2007)
and Diebold and Li (2006), and macroeconomic data from FRED. We compare the
different models by means of the conditional predictive ability test of Giacomini and
White (2006). We find that the yield curve has more forecast power for real variables
compared to inflation measures and that supervising the factor extraction for the
forecast target can improve the forecast performance of the factor model. We also
compare direct and indirect forecasts for the different models and find that the in-
direct forecasts perform better for our data and specification. We find that the yield
curve has forecast power for unemployment rate, real disposable income, and indus-
trial production. Similarly to Giacomini and Rossi (2006), Rudebusch and Williams
(2009), and Stock and Watson (1999) we find that the predictive ability of the yield
curve is somewhat unstable and has changed through the years. In particular, we
find the yield curve has more predictive power for the periods 1987-1994 (the early
Greenspan monetary regime) and 2006-2012 (the early Bernanke monetary regime)
as compared to the period 1994-2006 (the late Greenspan monetary regime).
In Chapter 3, we propose a continuous-time, Gaussian, linear state-space system
to model the relation between global mean sea level (GMSL) and the global mean
temperature (GMT) with the aim of making long term projections for the GMSL. We
provide a justification for the model specification based on popular semi-empirical
methods present in the literature and on zero-dimensional energy balance models.
We show that some of the models developed in the literature on semi-empirical mod-
els can be analysed within this framework. We use the sea-level data reconstruction
developed in Church and White (2011) and the temperature reconstruction from
Hansen, Ruedy, Sato, and Lo (2010). We compare the forecasting performance of
the proposed specification to the procedures developed in Rahmstorf (2007) and
Vermeer and Rahmstorf (2009). Finally, we compute projections for the sea level, con-
ditional on 21st century temperature scenarios, corresponding to the Special Report
on Emissions Scenarios (SRES) of the Intergovernmental Panel on Climate Change
(IPCC) fourth assessment report. Furthermore, we propose a bootstrap procedure to
compute confidence intervals for the projections, based on the method introduced
in Rodriguez and Ruiz (2009). We make projections of the sea-level rise from 2010
up to 2099. Across state-space specifications and temperature scenarios, we find as
best-case scenario an increase of the sea level of 0.09[m] and as worst-case scenario
an increase of 0.37[m] for the year 2099, with respect to the (smoothed) sea-level
value in 2009. This corresponds to an increase of roughly 0.14[m] (best case) and
0.42[m] (worst case) relative to the mean, smoothed 1990 sea-level value. The choice
of the trend component influences somewhat the forecast performance of the model.
CONTENTS ix
References
Church, J. A., White, N. J., 2011. Sea-level rise from the late 19th to the early 21st
century. Surveys in Geophysics 32 (4-5), 585–602.
Diebold, F. X., Li, C., 2006. Forecasting the term structure of government bond yields.
Journal of econometrics 130 (2), 337–364.
Giacomini, R., Rossi, B., 2006. How stable is the forecasting performance of the yield
curve for output growth? Oxford Bulletin of Economics and Statistics 68 (s1), 783–
795.
Giacomini, R., White, H., 2006. Tests of conditional predictive ability. Econometrica
74 (6), 1545–1578.
Gürkaynak, R. S., Sack, B., Wright, J. H., 2007. The us treasury yield curve: 1961 to the
present. Journal of Monetary Economics 54 (8), 2291–2304.
Hansen, J., Ruedy, R., Sato, M., Lo, K., 2010. Global surface temperature change.
Reviews of Geophysics 48 (4).
Nelson, C. R., Siegel, A. F., 1987. Parsimonious modeling of yield curves. Journal of
business, 473–489.
Rahmstorf, S., 2007. A semi-empirical approach to projecting future sea-level rise.
Science 315 (5810), 368–370.
Rodriguez, A., Ruiz, E., 2009. Bootstrap prediction intervals in state–space models.
Journal of time series analysis 30 (2), 167–178.
Rudebusch, G. D., Williams, J. C., 2009. Forecasting recessions: the puzzle of the
enduring power of the yield curve. Journal of Business & Economic Statistics 27 (4).
Stock, J. H., Watson, M. W., 1999. Business cycle fluctuations in us macroeconomic
time series. Handbook of macroeconomics 1, 3–64.
Vermeer, M., Rahmstorf, S., 2009. Global sea level linked to global temperature. Pro-
ceedings of the National Academy of Sciences 106 (51), 21527–21532.
DANISH SUMMARY
Denne afhandling indeholder tre selvstændige, uafhængige kapitler. Det statistiske
grundlag i alle tre kapitler er lineære, Gaussiske og state-space-systemer. I de første
to kapitler studerer vi en metode til at forudsige makroøkonomiske tidsserier og det
tredje er fokuseret på forudsigelse af det globale gennemsnitlige havniveau, betinget
af den globale gennemsnitstemperatur.
Mere konkret, så undersøger vi i Kapitel 1 prædiktionsevnen af en bestemt fak-
tormodel, hvor faktorerne er trukket ud fra et stort antal prædiktorer. Vi anvender en
semiparametrisk state-space-repræsentation af faktormodellen i hvilket prædiktions-
målet såvel som faktorerne er inkluderet i state-vektoren. Faktorerne er informerede
om prædiktionsmålet (superviseret) igennem state-ligningens dynamik. Baseret på
Kalman-rekursionerne fremsætter vi en metode til at bedømme bidraget fra prædik-
tionsmålet på de udtrukne faktorer. Vi prædikterer et mål ad gangen på baggrund af
de filtrerede states og estimerede parametre i state-space-systemet. Vi vurderer præ-
diktionsevnen af de foreslåede metoder uden for stikprøven i et simulationsstudie og
i en empirisk anvendelse på amerikanske makroøkonomiske data. Vi sammenligner
vores specifikation med andre multi- og univariate metoder og bruger Giacomini og
White (2006)’s test til at vurdere den relative betingede prædiktionsevne af model-
lerne. Vi vælger primært følgende konkurrerende modeller: principal components,
partiel mindste-kvadraters regression, ARMA(p,q) processer, en ikke-superviseret
faktormodel og en standard dynamisk faktormodel med separat prædiktions- og
state-ligning. Vi finder, at variablene som bidrager mest til variansen af de filtrerede
states, er de som drager størst fordel af det superviserede set up og vice versa. Den
foreslåede specifikation klarer sig specielt godt i forudsigelse af den amerikanske
centralbanksrente, arbejdsløshedsraten og den reelle disponible indkomst.
I Kapitel 2 studerer vi prædiktionsevnen af rentekurven for makroøkonomiske
tidsserier. Vi anvender et state-space-system, i hvilket prædiktionsobjektivet er in-
kluderet i state-vektoren. Dette resulterer i en udvidet dynamisk faktormodel i hvil-
ken faktorerne (niveau, hældning og krumning af rentekurven) er superviserede for
prædiktionsmålet. Faktorvægtene har struktur fra Nelson and Siegel (1987), og vi be-
tragter et prædiktionsmål ad gangen. Vi forudsiger forbrugerprisindekset, personlige
forbrugsudgifter, producentprisindekset, reel disponibel indkomst, arbejdsløsheds-
raten og industriel produktion. Alle tidsserier vedrører den amerikanske økonomi.
xi
xii CONTENTS
Vi sammenligner prædiktionspræstationen af vores specifikation til benchmark-
modeller såsom principal component regression, partiel mindste-kvadraters regres-
sion, ARMA(p,q) processer og totrins-prædiktionsmetoder baseret på faktormodeller.
Vi anvender rentekurvedata fra Gürkaynak, Sack og Wright (2007) og Diebold og Li
(2006) og makroøkonomisk data fra FRED. Vi sammenligner de forskellige modeller
ved hjælp af den betingede prædiktionsevnetest fra Giacomini og White (2006). Vi
finder, at rentekurven bedre prædikterer reelle variable, sammenlignet med infla-
tionsmål og at supervision af faktorudledningen kan forbedre prædiktionsevnen
af faktormodellen. Vi sammenligner også direkte og indirekte prædiktioner fra de
forskellige modeller og finder, at indirekte prædiktioner klarer sig bedre for vores data
og specifikationer. Vi finder, at rentekurven kan hjælpe med at prædiktere arbejdsløs-
hedsraten, reel disponibel indkomst og industriel produktion. Ligesom Giacomini
og Rossi (2006), Rudebusch og Williams (2009) og Stock og Watson (1999) finder vi,
at rentekurvens prædiktionsevne er noget ustabil og har ændret sig igennem årene.
I særdeleshed finder vi, at rentekurven har større prædiktionsevne for perioderne
1987-1994 (det tidlige Greenspan monetære regime) og 2006-2012 (det tidlige Bernan-
ke monetære regime), sammenlignet med 1994-2006 (det sene Greenspan monetære
regime).
I Kapitel 3 fremsætter vi et Gaussisk, lineært state-space-system i kontinuert
tid som model for relationen mellem det globale gennemsnitlige havniveau (GGH)
og den globale gennemsnitlige temperatur (GGT) med det formål at lave langsig-
tede forudsigelser af GGH. Vi giver en begrundelse for modelspecifikationen ba-
seret på populære semiempiriske metoder fra litteraturen samt nul-dimensionale
energibalance-modeller. Vi viser, at nogle af de semiempiriske metoder fra litteratu-
ren kan analyseres i dette set up. Vi bruger havniveaudata-rekronstruktion udviklet i
Church og White (2011) og temperatur-rekonstruktionen fra Hansen, Ruedy, Sato og
Lo (2010). Vi sammenligner prædiktionsevnerne af de foreslåede specifikationer til
procedurerne udviklet i Rahmstorf (2007) og Vermeer og Rahmstorf (2009). Endelig
udregner vi forudsigelserne for havniveauet betinget på det 21. århundredes SRES
temperaturscenarier fra IPCC’s fjerde vurderingsrapport. Vi fremsætter yderligere
en bootstrap-metode til at udregne konfidensintervaller for forudsigelserne base-
ret på metoden introduceret i Rodriguez og Ruiz (2009). Vi foretager forudsigelser
af havniveauet fra 2010 til og med 2099. På tværs af state-space-specifikationer og
temperaturscenearier finder vi, at i det bedste tilfælde vil der være en stigning i havni-
veauet på 0,09[m] og i det værste tilfælde vil stigningen være på 0,37[m] for året 2099,
mht. havniveauet i 2009. Dette svarer groft sagt til en stigning på mellem 0,14[m]
(bedste scenario) og 0,42[m] (værste scenario) relativt til det gennemsnitlige, udglat-
tede 1990-havniveau. Valget af trend-specifikationen påvirker prædiktionsevnen af
modellen i nogen grad.
CH
AP
TE
R
1SUPERVISION IN FACTOR MODELS USING A
LARGE NUMBER OF PREDICTORS
A STATE-SPACE APPROACH
Lorenzo Boldrini
Aarhus University and CREATES
Eric Hillebrand
Aarhus University and CREATES
1
2 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Abstract
In this paper we investigate the forecasting performance of a particular factor model
(FM) in which the factors are extracted from a large number of predictors. We use a
semi-parametric state-space representation of the FM in which the forecast objective,
as well as the factors, is included in the state vector. The factors are informed of
the forecast target (supervised) through the state equation dynamics. We propose
a way to assess the contribution of the forecast objective on the extracted factors
that exploits the Kalman filter recursions. We forecast one target at a time based on
the filtered states and estimated parameters of the state-space system. We assess the
out-of-sample forecast performance of the proposed method in a simulation study
and in an empirical application, comparing its forecasts to the ones delivered by
other popular multivariate and univariate approaches, e.g. a standard dynamic factor
model with separate forecast and state equations.
1.1. INTRODUCTION 3
1.1 Introduction
The availability of large datasets, the increase in computational power, and the ease
of implementation have made factor models an appealing tool in forecasting. Factor
models offer several advantages over other forecasting methods. For example, they do
not require the choice of the variables to include in the forecasting scheme (as struc-
tural models do), they make use of a large information set, they allow to concentrate
the information in all the candidate predictors in a relatively small number of factors,
and they can be estimated with simple and fast methods. Using many predictors
also allows to avoid the structural instability typical of low-dimensional systems. As
argued for instance in Stock and Watson (2006) and Stock and Watson (2002a), also
practitioners typically examine a large number of variables when making forecasts.
Forecasting using factor models is usually carried out in a two-step procedure, as
suggested for instance by Stock and Watson (2002b). In the first step the factors are
estimated using a set of predictors (that may include the lags of the forecast target)
and in a second step the estimated factors are used to forecast the target by means of
a forecast equation. In the two-step forecasting procedure suggested in Stock and
Watson (2002b) however, the same factors are used to forecast different targets. That
is, the selection of the factors is not supervised by the forecast target. In this paper we
study a method to supervise the factor extraction for the forecast objective in order to
improve on the predictive power of factor models. In the supervised framework, the
factors are informed of the forecast target (supervised) through the state equation
dynamics. Furthermore, we propose a way to assess the contribution of the forecast
objective on the extracted factors that exploits the Kalman filter recursions.
The forecasting properties of static, restricted, and general dynamic factor models
have been widely studied in the literature. Some examples are Boivin and Ng (2005)
and d’Agostino and Giannone (2012), who study the predictive power of different
approaches belonging to the class of general dynamic factor models. Alessi, Barigozzi,
and Capasso (2007), Stock and Watson (2002b), Stock and Watson (2002a), and Stock
and Watson (2006) compare the forecasting performance of factor models to dif-
ferent univariate and multivariate approaches. The evidence regarding the relative
merits of factor models in forecasting, compared to other methods, differs between
works. Stock and Watson (1999) and Stock and Watson (2002b) find a better forecast
performance of factor models compared to univariate methods for inflation and
industrial production, whereas Schumacher and Dreger (2002), Banerjee, Marcellino,
and Masten (2005), and Engel, Mark, and West (2012) find mixed evidence.
The latent factors in a FM can be estimated using principal components analysis
(PCA), as in Stock and Watson (2002a), by dynamic principal components analysis,
using frequency domain methods, as proposed by Forni, Hallin, Lippi, and Reichlin
(2000), or by Kalman filtering techniques. Comprehensive surveys on factor models
can be found in Bai and Ng (2008b), Breitung and Eickmeier (2006), and Stock and
Watson (2011).
4 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
In the standard approach to factor models, the extracted factors are the same for
all the forecast targets. One of the directions the literature has taken for improving
on this approach is to select factors based on their ability to forecast a specific target.
Different methods have been proposed in the literature that address this problem.
The method of partial least squares regression (PLSR), for instance, constructs a
set of linear combinations of the inputs (predictors and forecast target) for regres-
sion, for more details see for instance Friedman, Hastie, and Tibshirani (2001). Bai
and Ng (2008a) proposed performing PCA on a subset of the original predictors, se-
lected using thresholding rules. This approach is close to the supervised PCA method
proposed in Bair, Hastie, Paul, and Tibshirani (2006), that aims at finding linear com-
binations of the predictors that have high correlation with the target. In particular,
first a subset of the predictors is selected, based on the correlation with the target
(i.e. the regression coefficient exceeds a given threshold), then PCA is applied on the
resulting subset of variables. Bai and Ng (2009) consider ‘boosting’ (a procedure that
performs subset variable selection and coefficient shrinkage) as a methodology for
selecting the predictors in factor-augmented autoregressions. Finally, Giovannelli
and Proietti (2014) propose an operational supervised method that selects factors
based on their significance in the regression of the forecast target on the predictors.
The supervised dynamic factor model we study in this paper is based on a Gaus-
sian, factor-augmented, approximate, dynamic factor model in which the forecast
objective is modelled jointly with the factors. In this paper, by dynamic factor model
we mean a factor model in which the factors follow a dynamic equation. The system
has a linear state-space representation and we estimate it using maximum likelihood.
The likelihood function is delivered by the Kalman filter. Under this setup, we propose
a way to measure the contribution of the forecast objective on the extracted factors
that exploits the Kalman filter recursions. In particular, we compute the contribution
of the forecast target to the variance of the filtered factors and find a positive corre-
spondence between this quantity and the forecast performance of the supervised
scheme.
We assess the out-of-sample forecast performance of the supervised scheme by
means of a simulation study and in an empirical application. In the simulation study,
we vary the degree of correlation between the factors and forecast objective. We com-
pare the forecasts from the supervised model to two unsupervised FM specifications.
We find that the higher the correlation between factors and forecast target, the better
the forecasts of the supervised scheme. In the empirical application, we forecast
selected macroeconomic time series and compare the forecast performance of the
supervised FM to two unsupervised FM specifications and other multivariate and uni-
variate methods. We use the dataset from Jurado, Ludvigson, and Ng (2015), adding
two more variables: real disposable personal income and personal consumption
expenditure, excluding food and energy, and removing the index of aggregate weekly
Hours (BLS), because this series starts later than the others. The resulting dataset
1.2. FORECASTING WITH DYNAMIC FACTOR MODELS 5
comprises 132 variables. We forecast consumer price index (CPI), federal funds rate
(FFR), personal consumption expenditures deflator (PCEd), producer price index
(PPI), personal income (PEI), unemployment rate (UR), industrial production (IP),
real disposable income (RDI), and personal consumption expenditures (PCE). The
observations range from January 1960 to December 2011 and all variables refer to the
US economy.
The paper is organized as follows: in Section 1.2 we introduce the supervised
factor model and compare with other forecasting methods based on factor models;
in Section 1.3 we show how supervision can be measured using the Kalman filter
recursions; in Section 1.4 we provide some details on the computational aspects of
the analysis; in Sections 1.5 and 1.6 we describe the empirical application and the
simulation setup, respectively; finally, Section 1.7 concludes.
1.2 Forecasting with dynamic factor models
Let yt be the forecast objective, xt an N -dimensional vector of predictors (that may
or not include lags of the forecast objective), h the forecast horizon and T the last
available time-point in the estimation window.
Supervised factor model
We propose the following forecasting model. Consider the state-space system:[xt
yt
]=
[Λ 00 1
][ft
yt
]+
[εt
0
], εt ∼ N (0,H),[
ft+1
yt+1
]= c+T
[ft
yt
]+ηt , ηt ∼ N (0,Q), (1.1)
where ft ∈ Rk are latent factors, Λ is a matrix of factor loadings, T and c are a ma-
trix and a vector of coefficients, respectively, of suitable dimensions, εt ∈ RN and
ηt ∈Rk+1 are uncorrelated vectors of disturbances and H and Q are their respective
variance-covariance matrices. The forecast objective is placed in the state equation
together with the latent factors and the predictors are modelled in the measurement
equation. We consider joint estimation of the factors using the Kalman filter recur-
sions and maximum likelihood estimation for the parameters. The intuition behind
the model is that if the forecast objective is correlated with the factors, modelling
factors and forecast objective jointly should deliver a better estimate of the factors.
We define supervision to be the contribution of the forecast target to the estimation
of the latent factors. In the next section we derive the analytical expression of this
contribution and present a measure of supervision based on it.
The state equation can be understood as a factor augmented VAR (FAVAR), intro-
duced in Bernanke, Boivin, and Eliasz (2005), in which factors are included together
6 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
with observables in a VAR model. A specification similar to this one was used also in
Diebold, Rudebusch, and Boragan Aruoba (2006) to analyse the correlation between
the Nelson-Siegel factors and some macroeconomic variables.
We wish to extract factors from a large number of predictors and model them
jointly with the forecast objective. In order to find a parsimonious specification of
the factor model we select as factor loadings basis functions of RN . This corresponds
to taking a low order approximation of the vector of predictors at each point in
time. Virtually any basis of RN can be used. We choose discrete cosine basis for their
ease of implementation. Mallat (1999, Theorem 8.12) shows that a random vector in
CN can be decomposed into discrete cosine basis. In particular, any g ∈CN can be
decomposed into
gn = 2
N
N−1∑k=0
fnλk cos
[kπ
N
(n + 1
2
)],
for 0 ≤ n < N , where gn is the n − th component of g,
λk =2−1/2 if k = 0 and
1 otherwise
and
fn =⟨
gn ,λk cos
[kπ
N
(n + 1
2
)]⟩=λk
N−1∑n=0
gncos
[kπ
N
(n + 1
2
)],
are the discrete cosine transform of type I.
In our specification xt = gt + εt for each t = 1, ...,T and xt ,n = g t ,n + εt ,n with
n = 1, ..., N , where with xt we denote a vector of predictors. For each point in time we
then have g t ,n = 2N
∑N−1k=0 ft ,kλk cos
[kπN
(n + 1
2
)]. The weights ft ,k are estimated via
Kalman filter/smoother recursions. The cosine basis functions are then contained in
the factor loading matrix
Λ =
p2
N2N cos
[πN
(1+ 1
2
)]· · · 2
N cos
[(k−1)π
N
(1+ 1
2
)]...
......
p2
N2N cos
[πN
(N + 1
2
)]· · · 2
N cos
[(k−1)π
N
(N + 1
2
)] . (1.2)
The supervised factor model is then comprised of equations (1.1) and (1.2). The
forecasting scheme for this model is:
(i) estimation of the system parameters using maximum likelihood;
(ii) extraction of the factors using the Kalman filter;
1.2. FORECASTING WITH DYNAMIC FACTOR MODELS 7
(iii) the forecast yT+h is obtained as the last element of the vector[fT+h|TyT+h|T
]= T
h
[fT |TyT
]+
h−1∑i=0
Tic, (1.3)
where fT |T is the vector of filtered factors, h is the forecast lead, and T and c are
estimated parameters.
Note that the filtered and smoothed estimates for fT are the same.
Two-step procedure
Forecasting using dynamic factor models (DFM hereafter) is often carried out in a
two-step procedure as in Stock and Watson (2002a). Consider the model
yt+h = β(L)′ft +γ(L)yt +εt+h , (1.4)
xt ,i = λi (L)ft +ηt ,i , (1.5)
with i = 1, . . . , N and where ft = ( ft ,1, . . . , ft ,k ) are k latent factors, ηt = [ηt ,i , . . . ,ηt ,N ]′
and εt are idiosyncratic disturbances, β(L) = ∑qj=0β j+1L j , λi (L) = ∑p
j=0λi ( j+1)L j ,
and γ(L) = ∑sj=0γ j+1L j are finite lag polynomials in the lag operator L; β j ∈ Rk ,
γ j ∈ R, and λi j ∈ R are parameters and q, p, s ∈N0 are indices. The assumption on
the finiteness of the lag polynomials allows us to rewrite (1.4)-(1.5) as a static factor
model, i.e. a factor model in which the factors do not appear in lags:
yt+h = c +β′Ft +γ(L)yt +εt+h ,
xt = ΛFt +ηt , (1.6)
with Ft = [f′t , . . . , f′t−r ]′, r = max(q, p), the i -th row of Λ is [λi ,1, . . . ,λi ,r+1], and β =[β′
1, . . . ,β′r+1]′. The forecasting scheme is the following:
(i) extraction of the factors ft from the predictors xt , modelled in equation (1.5),
using either principal components analysis or the Kalman filter;
(ii) regression of the forecast objective on the lagged estimated factors and on its
lags, according to the forecasting equation (1.4);
(iii) the forecast is obtained from the estimated factors and regression coefficients
as
yT+h = c + β′Ft + γ(L)yT .
Stock and Watson (2002a) developed theoretical results for this two-step procedure,
in the case of principal components estimation. In particular, they show the asymp-
totic efficiency of the feasible forecasts and the consistency of the factor estimates.
8 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
The difference between the supervised DFM and the two-step forecasting proce-
dure is that in the former model the factors are extracted conditionally on the forecast
target. In the supervised framework the filtered/smoothed factors are tailored to the
forecast objective. Note that for a linear state-space system the Kalman filter delivers
the best linear predictions of the state vector, conditionally on the observations, for
a correctly specified model. Moreover, if the innovations are Gaussian, the filtered
states coincide with conditional expectations, for more details on the optimality
properties of the Kalman filter see Brockwell and Davis (2009).
1.3 Quantifying supervision
In this section we propose a statistic to quantify supervision. We are interested in
quantifying the influence of the forecast target on the filtered factors. To accomplish
this, we develop some results that hold for a general linear, state-space system with
non-random coefficient matrices. Consider the following state-space system:
yt = Ztαt +εt ,
αt+1 = Ttαt +Rtηt , (1.7)
where εt ∼W N (0,Ht ) and ηt ∼W N (0,Qt ) are uncorrelated random vectors, yt ∈RN ,
αt ∈Rk , and ηt ∈Rq , and the matrices Zt , Tt , and Rt are of suitable dimensions. Note
that in this context we assume white noise innovations.
In model (1.1), we include the observable forecast target as last element, both in
the measurement and in the state equation. In the notation of model (1.7), we are
therefore ultimately interested in the influence of the forecast target, the last element
in yt , denoted yt ,N (or yt in the notation of model (1.1)), on the filtered factors ft . To
be more precise, the objective is to quantify the influence of the sequence yi ,N i=1,...,t
(or yi i=1,...,t in the notation of model (1.1)) on ft , the filtered factors at time t .
The standard Kalman filter recursions (see for instance Durbin and Koopman
(2012)) for system (1.7) are:
vt = yt −Zt at ,
Ft = Zt Pt Z′t +Ht ,
Mt = Pt Z′t , (1.8)
at |t = at +Mt F−1t vt , Pt |t = Pt −Mt F−1
t M′t ,
at+1 = Tt at |t , Pt+1 = Tt Pt |t T′t +Rt Qt R′
t , (1.9)
for t = 1, . . . ,T , where Pt = E [(αt −at )(αt −at )], Pt |t = E[[αt −at |t ][αt −at |t ]′
], Ft =
E [vt v′t ], and at |t = Pt (αt ) = P (αt |y0, . . . ,yt ) and at = Pt−1(αt ) = P (αt |y0, . . . ,yt−1) are
the filtered state and one-step-ahead prediction of the state vector, respectively, and
Pt (·) is the best linear predictor operator, see Brockwell and Davis (2002) for more
1.3. QUANTIFYING SUPERVISION 9
details on the definition and properties of the best linear predictor operator. In the
particular case of Gaussian innovations in both the state and measurement equations,
the best linear predictor coincides with the conditional expectation. The forecasting
step in the Kalman recursions (1.9) can be written as
at+1 = Tt at +Kt vt
= Tt at +Kt (yt −Zt at )
= St at +Kt yt , (1.10)
with Kt = Tt Pt Z′t F−1
t and St = Tt −Kt Zt . Iterating backwards on the one-step-ahead
prediction of the state, the filtered state can be written as
at |t = at + Kt vt
= Nt at + Kt yt
= Nt(St−1at−1 +Kt−1yt−1
)+ Kt yt
= Nt
[St−1
(St−2at−2 +Kt−2yt−2
)+Kt−1yt−1
]+ Kt yt
= Nt[St−1St−2at−2 +St−1Kt−2yt−2 +Kt−1yt−1
]+ Kt yt
= . . .
= Nt
t−1∏i=1
St−i a1 +t−1∑i=1
(i−1∏`=1
St−`
)Kt−i yt−i
+ Kt yt , (1.11)
with Nt = (I− Kt Zt ), Kt = Pt Z′t F−1
t and the convention∏i−1`=1 St−` = Ik if i −1 < 1. The
contribution of the n-th observable on the filtered state at time t can be isolated from
the previous expression in the following way
at |t = Nt
t−1∏i=1
St−i a1 +t−1∑i=1
(i−1∏`=1
St−`
)Kt−i
N∑j=1
e j e′j yt−i
+ Kt
N∑j=1
e j e′j yt
= Nt
t−1∏i=1
St−i a1 +t−1∑i=1
(i−1∏`=1
St−`
)Kt−i
N∑j=1, j 6=n
e j e′j yt−i
+ Kt
N∑j=1, j 6=n
e j e′j yt
+ Nt
t−1∑i=1
(i−1∏`=1
St−`
)kt−i ,·n yt−i ,n
+ kt ,·n yt ,n , (1.12)
where with bt ,·n and bt ,n· we denote the n-th column and row of the matrix Bt , respec-
tively, and yt ,n is the n-th component of yt ; e j with j = 1, . . . , N are the canonical basis
vectors of RN . In (1.12) we made use of the identity∑N
j=1 e j e′j = IN . The contribution
of the n-th observable yi ,N i=1,...,t on the filtered state at |t is given by
snt = Nt
t−1∑i=1
(i−1∏`=1
St−`
)kt−i ,·n yt−i ,n + kt ,·n yt ,n . (1.13)
10 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
The first moment and the variance of snt are
E [snt ] = Nt
t−1∑i=1
(i−1∏`=1
St−`
)kt−i ,·nE [yt−i ,n]+ kt ,·nE [yt ,n],
var [snt ] = Nt
t−1∑i=1,i ′=1
(i−1∏`=1
St−`
)kt−i ,·ncov[yt−i ,n , yt−i ′,n]k′
t−i ′,·n
i ′−1∏`=1
S′t−`
N′t
+ kt ,·n var [yt ,n]k′t ,·n
+ 2Nt
t−1∑i=1
(i−1∏`=1
St−`
)kt−i ,·ncov[yt−i ,n , yt ,n]k
′t ,·n . (1.14)
Note that Ft = E [vt v′t ], Pt = E [(αt −at )(αt −at )′], and Kt = Pt Z′t F−1
t are non-random
matrices. If yt ,n is stationary with mean E [yt ,n] =µy.,n and autocovariance function
cov[yt ,n , yt−h,n] = γn(h), we can rewrite (1.14) as
E [snt ] =
Nt
t−1∑i=1
(i−1∏`=1
St−`
)kt−i ,·n + kt ,·n
µy.,n ,
var [snt ] = Nt
t−1∑i=1,i ′=1
(i−1∏`=1
St−`
)kt−i ,·nγn(i − i ′)k′
t−i ′,·n
i ′−1∏`=1
S′t−`
N′t
+ kt ,·nγn(0)k′t ,·n
+ 2Nt
t−1∑i=1
(i−1∏`=1
St−`
)kt−i ,·nγn(i )k
′t ,·n ,
(1.15)
or, in a more compact form
E [snt ] =
(Wt−1ιt−1 + kt ,·n
)µy.,n ,
var [snt ] = Wt−1Γ
nt−2W′
t−1
+ kt ,·nγn(0)k′t ,·n
+ 2Wt−1γnt−1k
′t ,·n ,
where Wt−1 =[
Nt kt−1,·n ,Nt
(∏1`=1 St−`
)kt−2,·n , . . . ,Nt
(∏t−2`=1 St−`
)k1,·n
], ιt−1 is a vec-
tor of ones of length t −1, γnt−1 =
[γn(1), . . . ,γn(t −1)
]′, and
Γnt−2 =
γn(0) γn(1) · · · γn(t −2)
γn(1) γn(0) · · · γn(t −3)...
.... . .
...
γn(t −2) γn(t −3) · · · γn(0)
. (1.16)
1.3. QUANTIFYING SUPERVISION 11
The distribution of the contribution of the n-th observable on the filtered state at
time t is
snt ∼ p
(E
[sn
t
], var
[sn
t
]), (1.17)
where p(µ,Σ) denotes the distribution function of yt ,n with mean µ and covariance
matrix Σ. The contribution of observable n on the j -th filtered state at time t is given
by snt , j = e′j sn
t with e j the j -th canonical basis vector of Rk .
Variance of filtered states explained by forecast objective
In this section we derive the variance ratio used as a measure of supervision. In
particular, we compute the fraction of the total variance of the filtered factors that is
explained by snt , the contribution of the forecast target. According to eqn. (1.11) and
assuming a1 to be a constant vector (typically a1 =µ= E [αt ] for a stationary system),
the variance of at |t can be written as
var [at |t ] = var
[Nt
t−1∑i=1
Bit yt−i
]+ var
[Kt yt
]+2cov
[Nt
t−1∑i=1
Bit yt−i , Kt yt
]
= Nt
t−1∑i=1, j=1
Bit cov
[yt−i ,yt− j
](B j
t )′N′t
+ Kt var[yt
]K′
t +2Nt
t−1∑i=1
Bit cov
[yt−i ,yt
]K′
t ,
where Bit =
(∏i−1`=1 St−`
)Kt−i and as in the previous section St = Tt −Kt Zt . If yt is
stationary, we can write
var[at |t
] = Nt
t−1∑i=1, j=1
BitΣ(i − j )(B j
t )′N′t
+ KtΣ(0)K′t +2Nt
t−1∑i=1
BitΣ(i )K′
t , (1.18)
where Σ(i − j ) = cov(yt−i ,yt− j ).
Notice that, since the sequence of filtered states depends on the initial values of
the filter, so does the sequence sni i=1,...,t . As a consequence, it is not a stationary
and ergodic sequence. Its variance changes in time and in order to estimate it, we
first need to estimate the autocovariance function of the sequence of observations
yi i=1,...,t and the parameters of the system and then evaluate expressions (1.16) and
(1.18).
The variance of the j -th filtered factor explained by the n-th variable can then be
12 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
assessed by means of the ratio
r j ,nt =
e′j var[
snt
]e j
e′j var[at |t
]e j
, (1.19)
where e j is the j -th canonical basis vector of Rk , as before. This quantity can be
estimated by consistently replacing the data-generating parameters with consistent
estimates and the autocovariances of yt by their sample counterparts (under the con-
dition of ergodic stationarity). Note that the variance ratio has the same expression
also when adding a constant ct to the state equation.
1.4 Computational aspects
The objective of this study is to determine the forecasting power of the supervised fac-
tor model (1.1)-(1.2). The forecast performance is based on out-of-sample forecasts
for which a rolling window of fixed size is used for the estimation of the parameters.
The log-likelihood is maximized for each estimation window.
Estimation method
The parameters of the state-space model are estimated by maximum likelihood. The
likelihood is delivered by the Kalman filter. We employ the univariate Kalman filter
derived in Koopman and Durbin (2000) as we assume a diagonal covariance matrix
for the innovations in the measurement equation. The maximum of the likelihood
function has no explicit form solution and numerical methods have to be employed.
We make use of the following two algorithms.
• CMA-ES. Covariance Matrix Adaptation Evolution Strategy, see Hansen and
Ostermeier (1996)1. This is a genetic algorithm that samples the parameter
space according to a Gaussian search distribution which changes according to
where the best solutions are found in the parameter space;
• BFGS. Broyden-Fletcher-Goldfarb-Shanno, see for instance Press, Teukolsky,
Vetterling, and Flannery (2002). This algorithm belongs to the class of quasi-
Newton methods and requires the computation of the gradient of the function
to be minimized.
The CMA-ES algorithm performs very well when no good initial values are available
but it is slower to converge than the BFGS routine. The BFGS algorithm, on the
other hand, requires good initial values but converges considerably faster than the
CMA-ES algorithm (once good initial values have been obtained). Hence, we use
1See https://www.lri.fr/~hansen/cmaesintro.html for references and source codes. The au-thors provide C source code for the algorithm which can be easily converted into C++ code.
1.5. EMPIRICAL APPLICATION 13
the CMA-ES algorithm to find good initial values and then the BFGS one to perform
the minimizations with the different rolling windows of data. We use algorithmic (or
automatic) differentiation2 to compute gradients. We make use of the ADEPT C++
library, see Hogan (2013)3. The advantage of using algorithmic differentiation over
finite differences is twofold: increased speed and elimination of approximation errors
in the computation of the gradient.
Speed improvements
To gain speed we chose C++ as the programming language, using routines from the
Numerical Recipes, Press et al. (2002) 4. We compile and run the executables on a
Linux 64-bit operating system using the GCC compiler 5. We use Open MPI 1.6.4
(Message Passing Interface) with the Open MPI C++ wrapper compiler mpic++ to
parallelise the maximum likelihood estimations 6. We compute gradients using the
ADEPT library for algorithmic differentiation, see Hogan (2013).
1.5 Empirical application
We wish to assess the forecasting performance of model (1.1)-(1.2). We fix the number
of latent factors at 1, 2, and 3 for the models involving factors7. The complete sample
size is T = 617, the rolling window for the parameter estimation has size R = 306, and
the number of forecasts is S = 300.
Data
We use the Jurado, Ludvigson and Ng dataset as used in Jurado et al. (2015) adding
two more variables, namely real disposable income (RDI) and personal consumption
expenditure excluding food and energy (PCE) and removing the Index of Aggregate
Weekly Hours (BLS). The resulting dataset comprises 132 variables. We have applied
the same transformations as in Jurado et al. (2015) to achieve stationarity for the
series in common with this dataset. For RDI and PCE we used the same transforma-
tions used for the personal income (PI) and for the personal consumption deflator
(PCEd), respectively. Details on the Jurado, Ludvigson and Ng dataset used in Ju-
rado et al. (2015) are provided by the authors at http://www.econ.nyu.edu/user/
2See for instance Verma (2000) for an introduction to algorithmic differentiation.3For a user guide see http://www.cloud-net.org/~clouds/adept/adept_documentation.
pdf.4See Aruoba and Fernández-Villaverde (2014) for a comparison of different programming languages
in economics and Fog (2006) for many suggestions on how to optimize software in C++.5See http://gcc.gnu.org/onlinedocs/ for more information on the Gnu Compiler Collection,
GCC.6See http://www.open-mpi.org/ for more details on Open MPI and Karniadakis (2003) for a review
of parallel scientific computing in C++ and MPI.7See below for more details on the choice of the number of factors.
14 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
ludvigsons/data.htm.
The details for the time series added to the Jurado, Ludvigson and Ng dataset are
the following:
• PCE. Series ID: DPCCRC1M027SBEA, Title: Personal consumption expendi-
tures excluding food and energy, Source: U.S. Department of Commerce: Bu-
reau of Economic Analysis, Release: Personal Income and Outlays, Units: Bil-
lions of Dollars, Frequency: Monthly, Seasonal Adjustment: Seasonally Adjusted
Annual Rate, Notes: BEA Account Code: DPCCRC1, For more information about
this series see http://www.bea.gov/national/.
• RDI. Series ID: DSPIC96, Title: Real Disposable Personal Income, Source: U.S.
Department of Commerce: Bureau of Economic Analysis, Release: Personal In-
come and Outlays, Units: Billions of Chained 2009 Dollars, Frequency: Monthly,
Seasonal Adjustment: Seasonally Adjusted Annual Rate, Notes: BEA Account,
Code: A067RX1, A Guide to the National Income and Product Accounts of
the United States (NIPA) - (http://www.bea.gov/national/pdf/nipaguid.
pdf).
The RDI and PCE series have been taken from the FRED (Federal Reserve Economic
Data) database and can be downloaded from the website of the Federal Reserve Bank
of St. Louis: http://research.stlouisfed.org/fred2, Help: http://research.
stlouisfed.org/fred2/help-faq.
The macroeconomic variables selected as forecast objectives are: consumer price
index (CPI), federal funds rate (FFR), personal consumption expenditures deflator
(PCEd), producer price index (PPI), personal income (PEI), unemployment rate (UR),
industrial production (IP), real disposable income (RDI), and personal consumer
expenditures (PCE).
The observations in levels range from January 1960 to December 2011 for a total
of 624 observations, and from March 1960 to December 2011 after being transformed
to stationarity, for a total of 622 data points. The data refers to the US economy.
Selection of number of factors
The selection of the number of factors is a key aspect in dynamic/static factor mod-
els. A widely used information-criterion-based method for static factor models was
derived by Bai and Ng (2002). Under appropriate assumptions, they show that their
method can consistently identify the number of factors as both the cross-section and
the sample-size tend to infinity. The method was extended to the case of restricted
dynamic models by Bai and Ng (2007) and Amengual and Watson (2007). The Bai
and Ng (2002) criterion was found to overestimate the true number of factors in
simulation studies by e.g. Hallin and Liška (2007), who propose a new method, valid
under more general assumptions, that exploits the properties of the eigenvalues of
1.5. EMPIRICAL APPLICATION 15
sample spectral density matrices. Alessi, Barigozzi, and Capasso (2010) follow the idea
of Hallin and Liška (2007) to improve on Bai and Ng (2002) in the less general case of
static factor models. They show using simulations that their method performs well, in
particular under large idiosyncratic disturbances. Another method for selecting the
number of factors in static approximate factor models and based on the eigenvalues
of the variance-covariance matrix of the panel of data, was proposed by Ahn and
Horenstein (2013).
We find that the Alessi et al. (2010) test is somewhat dependent on the number
and sizes of the subsamples required by the test. Similarly, the number of factors
selected using the Ahn and Horenstein (2013) eigenvalue ratio test, is somewhat
sensitive to the choice of the maximum allowed number of factors. Motivated by
this and by the empirical finding that models using a low number of factors tend to
forecast better (see e.g. Stock and Watson (2002b) for the case of output and inflation)
in this work we consider models with a fixed, low number of factors. In particular, we
consider factor models with 1, 2, and 3 factors. Increasing the number of factors was
seen not to further improve the forecasts.
Competing models
We choose different competing models widely used in the forecasting literature in
order to assess the relative forecasting performance of the supervised DFM. We divide
these models into direct multi-step and indirect (recursive) forecasting models. In
the factor models considered as well as in the principal components regressions
and partial least squares regressions we extract 1, 2, and 3 factors. In the following
we denote with h the forecast horizon, yt the forecast objective, xt = [x1t , . . . , xN
t ] an
(N ×1) vector of predictors, εt a Gaussian white noise innovation, ft = [ f 1t , . . . , f k
t ] a
(k ×1) vector of factors andΛ a matrix of factor loadings.
Direct forecasting models
The first model is the following restricted AR(p) process
yt+h = c +φ1 yt + . . .φp yt−p +εt+h . (1.20)
The second model is a restricted M A(q) process
yt+h = c +θ1εt + . . .θqεt−q +εt+h . (1.21)
Both models are estimated by maximum likelihood. The lags p and q are selected
for each estimation sample as the values that minimize the Bayesian information
criterion. In particular, we consider p, q ∈ 1,2,3.
The third model is principal component regression (PCR). In the first step, prin-
cipal components are extracted from the regressors Xt = [x1t , . . . , xN
t , yt ]; yt+h is then
regressed on them to obtain βPC R for time indexes 1 ≤ t ≤ Ti −h. In the second step,
16 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
the principal components are projected at time Ti and then multiplied by βPC R to
obtain the h-period ahead forecast.
The fourth model considered is partial least squares regression (PLSR). In the
first step, the partial least squares components ymt are computed using the forecast
target yt : h ≤ t ≤ Ti and the predictors Xt = [x1t , . . . , xN
t , yt ] with 1 ≤ t ≤ Ti −h where
M ≤ (N +1) is the number of partial least squares components and N +1 is the num-
ber of predictors, including the lagged value of the forecast objective. In the second
step, the partial least squares components ymt are regressed on the predictors Xt to
recover the coefficient vector βPLSR . Note that as the partial least squares compo-
nents are a linear combination of the regressors, the relation is exact, i.e. the residuals
from this regression are (algebraically) null. In the third step, the partial least squares
components are projected at time Ti by multiplying YTi by βPLSR . The projected
PLSR components at time Ti are then summed to obtain the h-period ahead forecast
yTi+h =∑Mm=1 ym
Ti.
The fifth direct forecasting method considered is a two-step procedure as de-
scribed in equations (1.4).
The sixth direct forecasting method is the one based on the principal components
estimation approach in Stock and Watson (2002a) to a specific version of model (1.6).
In particular we take as forecasting equation
yt+h = c +β′ft + yt +εt+h ,
xt = Λft +ηt , (1.22)
with ft = [ f 1t , . . . , f k
t ]. The predictors are in xt and are standardized for each forecasting
window. As described in Stock and Watson (2002a) the factor loadings Λ and the
factors ft for t = 1, . . . ,T can be estimated using principal components. Denote with
F = [f1, . . . , fT ] and X′ = [x1, . . . ,xT ]; the estimator of the loadings Λ is the matrix made
of the eigenvectors corresponding to the largest eigenvalues of the matrix X′X and the
factors are estimated by F = XΛ. In the present paper, we are focusing on forecasting
and hence any estimated rotation of the factors will suffice for the analysis.
Indirect forecasting models
The first model is the following AR(p) process
yt+h = c +φ1 yt+h−1 + . . .φp yt+h−p +εt+h . (1.23)
The second model is a M A(q) process
yt+h = c +θ1εt+h−1 + . . .θqεt+h−q +εt+h . (1.24)
Both models are estimated by maximum likelihood. The lags p and q are selected
for each estimation sample as the values that minimize the Bayesian information
criterion. In particular, we consider p, q ∈ 1,2,3.
1.5. EMPIRICAL APPLICATION 17
The third indirect forecasting method is an alternative two-step procedure. We
specify a dynamic equation for the factors and a static one between the factors and
the predictors/forecast target. In particular, we allow the factors Ft ∈Rk to follow the
autoregressive dynamics:
Ft+1 = c+TFt +νt , (1.25)
where T and c are a matrix and a vector of coefficients, respectively, and νt is a vector
of disturbances with E [νtν′t ] =Σ, and a static equation is specified for the mapping
between the factors and the predictors/forecast target, such as[xt
yt
]= ZFt +εt , (1.26)
where Z is a matrix of factor loadings and εt is an innovation vector, with E [εtε′t ] =Ω.
In particular, we use the factor loading matrix (1.2). Forecasts can be constructed by
estimating the system, iterating on the factor equation and then mapping the factors
to the forecast objective using the estimated factor loadings. Assuming Gaussianity
of the idiosyncratic errors, for instance, the system (1.25-1.26) can be estimated
maximizing the likelihood delivered by the Kalman filter. In this case a forecasting
scheme would be of this type:
(i) estimation of the system parameters by maximum likelihood;
(ii) extraction of the factors using the Kalman filter;
(iii) forecasting of factors using the state equation
fT+h =[
Th
fT +h−1∑i=0
Tic
], (1.27)
where ft represent estimated factors;
(iv) the forecast is then the last element of the vector[xT+h
yT+h
]= ZfT+h . (1.28)
Finally, we compare the forecast performance of the supervised model (1.1-1.2) to its
unsupervised counterpart. Namely, in this specification the factors are first extracted
using the Kalman filter and the forecast are then obtained using the forecast equation
yt+h = c + f′tβ+γyt +ut , (1.29)
where c, β, and γ are parameters to be estimated, ut is the error term, and ft is the
vector of filtered factors.
18 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Forecasting
Forecasting scheme
The aim is to compute the forecast of the objective variable yt at time t +h, i.e. yt+h ,
where h is the forecast lead. We consider a rolling windows scheme. The reason is
that one of the requirements for the application of the Giacomini and White (2006)
test, in case of nested models, is to use rolling windows. The variables, including the
forecast target, are made stationary according to the transformations used in Jurado
et al. (2015). We standardize the variables in the estimation windows by subtracting
the time average and dividing by the standard deviation.
We build series of forecast errors of length S for all forecast targets. The com-
plete time series is indexed Yt : t ∈ N>0, t ≤ T where T is the sample length of
the complete dataset and Yt = x1t , ..., xN
t , yt . The estimation sample takes into ac-
count observations indexed Yt : t ∈ N>0,Ti −R +1 ≤ t ≤ Ti for i ∈ N>0, i ≤ S with
T1 = R = T ∗−S −hmax +1 the index of the last observation of the first estimation
sample, which coincides with the size of the rolling window, and Ti = T1 + i for
i ∈N>0, i ≤ S and hmax is the maximum forecast lead. The forecasting strategy for
h-step ahead forecasts for the supervised factor model (1.1)-(1.2) is the following (for
the competing models the forecasting scheme is analogous), for i = 1, . . . ,S:
(i) estimation of the system parameters using information from time Ti −R +1 up
to time Ti by maximizing the log-likelihood function delivered by the Kalman
filter;
(ii) computation of the filtered state vector at time Ti , i.e. αTi |Ti (note that the last
element of αTi |Ti is yTi );
(iii) the forecast is then:
yTi+h|Ti = [01×L : 1]
[T
hαTi |Ti +
h−1∑i=0
Tic
], (1.30)
where the parameter matrices are relative to equation (1.1).
The forecasting scheme for the competing methods is analogous.
In particular, the complete sample size is T = 622, the rolling window has size
R = 311, and the number of forecasts is S = 300. The 1-step ahead forecasts range
from February 1986 to January 2011. The 12-step ahead forecasts range from January
1987 to December 2011.
1.5. EMPIRICAL APPLICATION 19
Test of forecast performance
We make use of the conditional predictive ability test proposed in Giacomini and
White (2006)8 to assess the forecasting performance of the supervised factor model
(1.1)-(1.2), relative to the other forecasting methods. In particular, we use a quadratic
loss function. This test is valid also when comparing nested models, provided a rolling
scheme for parameter estimation is used. The autocorrelations of the loss differentials
are taken into account by computing Newey and West (1987) standard errors. We
follow the “rule of thumb” in Clark and McCracken (2011) and take a sample split
ratio π= SR approximately equal to one.
Empirical application results
In this subsection we present results corresponding to the empirical application. The
mean square prediction error ratios between forecasts from the supervised model
and the competing models can be found in tables 1.1-1.9 in Appendix 1.10. The
supervised factor model corresponds to equations (1.1) with discrete cosine basis as
loadings, equation (1.2). In the tables, three, two, and one stars refer to significance
levels 0.01, 0.05, and 0.10 for the null hypothesis of equal conditional predictive ability
for the Giacomini and White (2006) test. The different forecasting models are labelled
according to the following convention:
• model 1. Principal component regression (PCR);
• model 2. Partial least squares regression (PLSR);
• model 3. AR(p) direct, eqn. (1.20);
• model 4. MA(q) direct, eqn. (1.21);
• model 5. AR(p) indirect, eqn. (1.23);
• model 6. MA(q) indirect, eqn. (1.24);
• model 7. Stock and Watson two-step procedure, eqn. (1.22);
• model 8. Unsupervised factor model (1.25), and (1.26) with discrete cosine
basis factor loadings, eqn. (1.2);
• model 9. Unsupervised factor model as in Section 1.2 with discrete cosine basis
factor loadings, eqn. (1.2);
• model 10. Supervised factor model as in eqn. (1.1) with discrete cosine basis
factor loadings (1.2).
8The authors provide MATLAB codes at http://www.runshare.org/CompanionSite/site.do?siteId=116 for the test.
20 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
For reasons explained in Section 1.5, we estimate the supervised and unsupervised
factor models using 1, 2, and 3 factors.
Looking at the tables 1.1-1.9, we can make the following remarks (divided with
respect to the different number of factors used):
(i) 1 factor. The supervised factor model, eqn. (1.1), in general delivers forecasts
better than or similar to the other forecasting methods. In more than 56% of the
cases the model performs better than the competing ones, in 23% equally well
and in roughly 20% of the cases it performs worse. However, of the 56% cases
in which the model performs better, 37% of them are statistically significant at
theα= 10% significance level, whereas of the 17% of cases in which it performs
worse, only 9% are statistically significant at the α = 10% significance level.
The supervised factor model offers better forecasts relative to unsupervised
ones for most targets. The model forecasts particularly well the federal funds
rate (FFR). The improvements over unsupervised factor models 7, 8, and 9, are
particularly marked for this variable;
(ii) 2 factors. In most cases the supervised factor model delivers forecasts similar
to or better than the other methods. In more than 51% of the cases the model
performs better than the competing ones, in 36% of which the differences are
statistically significant at α= 10% significance level, in 23% equally well and in
roughly 26% of the cases it performs worse, in 15% of which the differences are
statistically significant at the α= 10% significance level. The supervised factor
model forecasts particularly well the federal funds rate (FFR);
(iii) 3 factors. In most cases the supervised factor model delivers forecasts similar
to or better than the other methods. In more than 60% of the cases the model
performs better than the competing ones, in 33% of which the differences
are statistically significant at the α = 10% significance level, in 20% equally
well and in roughly 20% of the cases it performs worse, in 17% of which the
differences are statistically significant at the α = 10% significance level. The
supervised factor model forecasts particularly well the unemployment rate
(UR), personal income (PEI), and real disposable income (RDI). Improvements
over the unsupervised models 7, 8, and 9 are particularly clear for UR and RDI.
The indirect MA(q) process is hard to beat in forecasting inflation measures and
the federal funds rate at lead h = 1. For the rest of variables/leads the supervised
factor model performs well. The supervised factor model (model 10) performs well
in forecasting unemployment rate, real disposable income, and the federal funds
rate. These findings are somewhat similar to the ones in Stock and Watson (2002b),
in which it was found that the factors have more predictive power for real variables
rather than for inflation measures. The results are robust to the choice of sample split
as can be seen from tables 1.1-1.9.
1.6. SIMULATIONS 21
In table 1.10, in Appendix 1.10, are reported the ratios between the variance of the
contribution to the filtered factors of the forecast target and the total variance of the
filtered factors, equation (1.19), for all variables. We notice a positive relation between
the value of this ratio and the forecast performance of the supervised factor model.
For example CPI has a much lower impact on the filtered factors compared to FFR
and UR. A possible interpretation is that the forecast objectives which influence more
the extraction of the unobserved factors benefit more from the supervised framework.
This suggests that the supervised factors may contain additional information with
respect to unsupervised ones.
From tables 1.1-1.9 it remains unclear what the best number of factors is, in terms
of forecasting performance. The best number of factors seems to change with the
forecast target and sample split.
1.6 Simulations
We perform two simulation experiments according to two different data generating
processes (hereafter DGPs). We simulate a state-space system according to equations
(1.1) with different loading coefficients:
case 1 discrete cosine basis as loadings, equation (1.2);
case 2 random loadings, generated as independent draws from a normal distri-
bution N (0,1).
In both cases, the state vector follows a three dimensional, stable VAR(1). The first two
components of the state vector ft ,1 and ft ,2, are treated as latent factors whereas the
third component ft ,3, is regarded as the forecast objective. We simulate the system
under different correlations between the factors and the forecast objective, namely
ρ f1,y and ρ f2,y , by restricting the unconditional variance-covariance matrix of the
state vector. For pairs of indexes i ; j = 1;2 and i ; j = 2;1 we fix the correlations
ρ fi ,y = 0.5 and ρ fi , f j = 0.1 and let ρ f j ,y ∈ 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9. We then
compute forecasts using models 8, 9, and 10 previously defined and reported again
here for convenience:
model 8. Unsupervised factor model (1.25) and (1.26) with discrete cosine basis
factor loadings, eqn. (1.2);
model 9. Unsupervised factor model as in Section 1.2 with discrete cosine basis
factor loadings, eqn. (1.2);
model 10. Supervised factor model as in eqn. (1.1) with discrete cosine basis
factor loadings (1.2).
The complete sample size in the simulations is T = 600, the rolling window has size
R = 289, and the number of forecasts is S = 300.
22 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Simulations results
In this subsection we present results relative to the simulation exercise. The mean
square prediction error ratios between forecasts from the supervised model and the
competing models can be found in tables 1.11-1.12 in Appendix 1.10. The supervised
factor model corresponds to equations (1.1) with discrete cosine basis as loadings,
equation (1.2). Looking at tables 1.11 and 1.12, we can make the following remarks
divided according to the DGP and correlations.
I discrete cosine basis loadings
(i) varying ρ f1,y and fixed ρ f2,y . In around 50% of the cases the supervised
factor model performs better than the unsupervised counterparts (in
34% of which the difference is statistically significant at the α = 0.10
significance level), in 30% of the cases it delivers the same forecasting
performance as the other two methods and in the remaining 20% of
cases it delivers slightly worse forecasts (in 11% of which the difference is
statistically significant at the α= 0.10 significance level);
(ii) varying ρ f2,y and fixed ρ f1,y . In around 60% of the cases the supervised
factor model performs better than the unsupervised counterparts (in 30%
of which the difference is statistically significant at the α = 0.10 signifi-
cance level), in 25% of cases it delivers the same forecast performance
as the other two methods and in the remaining 15% of cases it delivers
slightly worse forecasts (in 21% of which the difference is statistically
significant at the α= 0.10 significance level).
II random loadings
(i) varying ρ f1,y and fixed ρ f2,y . In around 58% of the cases the supervised
factor model performs better than the unsupervised counterparts (in 34%
of which the difference is statistically significant at the α = 0.10 signifi-
cance level), in 19% of cases it delivers the same forecast performance
as the other two methods and in the remaining 23% of cases it delivers
slightly worse forecasts (in 42% of which the difference is statistically
significant at the α= 0.10 significance level);
(ii) varying ρ f2,y and fixed ρ f1,y . In around 75% of the cases the supervised
factor model performs better than the unsupervised counterparts (in 28%
of which the difference is statistically significant at the α = 0.10 signifi-
cance level), in 15% of cases it delivers the same forecast performance as
the other two methods and in the remaining 10% of cases it delivers slighly
worse forecasts (in 11% of which the difference is statistically significant
at the α= 0.10 significance level).
1.7. CONCLUSIONS 23
Furthermore, we notice that even with moderate levels of correlation between the
forecast objective and the factors, the supervised specification delivers on average
better forecasts with respect to the unsupervised counterparts.
1.7 Conclusions
In this paper we study the forecasting properties of a supervised factor model. In this
framework the factors are extracted conditionally on the forecast target. The model
has a linear state-space representation and standard Kalman filtering techniques
can be used. Under this setup, we propose a way to measure the contribution of the
forecast objective on the extracted factors that exploits the Kalman filter recursions.
In particular, we compute the contribution of the forecast target to the variance of
the filtered factors and find a positive correspondence between this quantity and the
forecast performance of the supervised scheme.
We assess the forecast performance of the supervised factor model with a simula-
tion study and an empirical application. The simulated data are generated according
to different levels of correlation between the forecast objective and the factors. In
the simulations experiment, we find that if the forecast objective is correlated with
the factors the supervised factor model improves, on average, forecast performance
compared to unsupervised schemes.
In the empirical application the supervised FM is used to forecast macroeconomic
variables using factors extracted from a large number of predictors. The macroeco-
nomic data are taken from the Jurado, Ludvigson and Ng dataset and FRED. We
estimate the model considering one, two, and three factors. We forecast consumer
price index (CPI), the federal funds rate (FFR), personal consumption expenditures
deflator (PCEd), the producer price index (PPI), personal income (PEI), the unem-
ployment rate (UR), industrial production (IP), the real disposable income (RDI), and
personal consumption expenditures (PCE) relative to the US economy.
We find that supervising the factor extraction can improve forecasting perfor-
mance compared to unsupervised factor models and other popular multivariate and
univariate forecasting models. For this dataset and specification the supervised factor
model outperforms partial least squares regressions and principal components re-
gressions on most targets. In forecasting inflation, both measured by consumer price
index and producer price index, M A(q) processes are difficult to beat whereas the
supervised factor model performs particularly well in forecasting the federal funds
rate, the unemployment rate, and real disposable income. These findings are similar
to the ones in Stock and Watson (2002b), in which it was found that the factors have
more predictive power for real variables rather than for inflation measures.
We find that variables which contribute more to the variance of the filtered states,
i.e. a higher r j ,nt , equation (1.19), are the ones which benefit more from the supervised
framework and vice versa. Furthermore, supervising the factor extraction leads in
24 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
most cases to improved forecasts, compared to unsupervised two-step forecasting
schemes.
1.8 Acknowledgements
This research was supported by the European Research Executive Agency in the
Marie-Sklodowska-Curie program under grant number 333701-SFM.
1.9. REFERENCES 25
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28 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
1.10 Appendix
Tables
In this section we report mean square forecast errors (MSFE) ratios corresponding to
the empirical application (see Section 1.5) and the simulation exercise (see Section
1.6). The results relative to the empirical application correspond to MSFE ratios
between model 10 and the competing models (see Section 1.5 for the description
of the different models involved) and are contained in tables 1.1-1.9. We consider
different subsamples of the dataset and estimate the factor models using 1, 2, and
3 factors. The results relative to the simulation exercise correspond to MSFE ratios
between model 10 and models 8 and 9 and are contained in tables 1.11-1.12. We
estimate the factor models using 2 factors. In the tables below, three, two, and one
stars refer to significance levels 0.01, 0.05, and 0.10 for the null hypothesis of equal
conditional predictive ability for the Giacomini and White (2006) test. In table 1.10
are reported the ratios between the variance of the contribution to the filtered factors
of the forecast target and the total variance of the filtered factors, equation (1.19), for
all variables.
1.10. APPENDIX 29
Table 1.1. MSFE ratios for whole forecast sample (1 factor).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 0,98 1,04 1,11 1,34* 1,11 1,34* 0,99 0,94 0,993 0,98 0,9 0,99 0,99 0,96 1,01 0,99 0,99 0,99
CPI 6 0,99 0,82** 0,99 0,95 1 1 0,99 1 0,999 0,98 0,79** 1,01 1,01 1 1 0,98 1 0,9812 1,02 0,83*** 1,01 1,01 1 1 1,02 1 1,02
1 0,78*** 0,47*** 0,95 1 0,95 1 1,11 1,03 0,98***3 0,75*** 0,32*** 0,68*** 0,65*** 0,73*** 0,79** 0,82 0,94 0,85**
FFR 6 0,72*** 0,26*** 0,89** 0,87*** 0,94* 0,92** 0,83 0,97 0,85***9 0,86 0,37*** 1,02 1,03 0,97*** 0,98*** 0,98 0,99 112 0,75*** 0,37*** 0,91** 0,92* 0,99 0,99 0,87 1 0,89***
1 0,94** 0,99 1,15 1,31* 1,15 1,31* 0,99 0,91* 13 1 0,91 0,99 0,99 0,98 1 0,99 0,98 0,99
PCEd 6 0,99 0,83*** 0,99 0,99 0,99* 1 0,99 1 0,999 0,99 0,87 1,01 1 1 1 1 1 112 1 0,85*** 0,97 0,97** 1 1 0,98 1 0,97
1 0,91** 0,88 1,1 1,27** 1,1 1,27** 1 0,79*** 13 0,98 0,85*** 0,97 0,97 0,98 1 0,98 0,98 0,98
PPI 6 0,99 0,82 1 1 1,01 1 1 1 19 0,99 0,79 0,97 0,97 1 1 0,98 1 0,9812 1,04 0,9* 1,05 1,03 1 1 1 1 1
1 1,03 0,95 0,94* 0,94** 0,94* 0,94** 1,04 1,03 13 1,02 0,92 0,96** 0,95** 0,97* 0,97* 1,03 1,01 1,02
PEI 6 1,01 0,89* 0,99*** 0,99*** 0,99** 0,99** 1,02 1 1,029 1 0,91** 0,97 0,96 1 1 1,01 1 1,0112 0,99 0,84 0,97 0,98 1 1 0,98 1 0,97
1 0,97 1,02 0,9 0,85* 0,9 0,85* 1,06 0,66*** 13 1,04 1,01 0,99 0,94** 0,99 0,92** 1,05 0,77*** 1,06
UR 6 1,05 0,83 0,96 0,99 0,99 0,95** 1,05 0,92** 1,059 1,03 0,89 0,95 1 0,99 0,98* 0,99 0,97* 1,0112 0,98*** 0,82 0,96** 0,95** 0,99 1** 0,94 0,99** 0,95**
1 0,92* 0,99 0,9 0,87 0,9 0,87 0,97 0,94 0,99***3 1,05 0,98 1,08 1,07 1,06 0,97 1,04 0,96 1,02
IP 6 1,02 0,93 1,01 1,02 1 0,98 0,98 0,99 0,999 0,96 0,89* 0,99 0,98 0,99 0,99 0,95 0,99 0,95**12 0,97 0,82** 0,97** 0,98*** 0,99 1 0,95 1 0,95
1 0,96 0,8*** 0,99 1,01 0,99 1,01 1,01 0,95 13 0,99 0,82*** 0,99 0,99 0,98 0,99 1 1 1*
RDI 6 0,99 0,78** 0,99 0,99 1 1 1 1 19 0,99 0,87*** 1 1 1 1 1,01 1 1,0112 1 0,85*** 1 1 1 1 1 1 1
1 0,63*** 1,01 1,38** 1,72*** 1,38** 1,72*** 0,99 0,58*** 13 1,05 0,84** 1,04 1,04 0,99 1,06 1,03 1,05 1,03
PCE 6 0,99 0,75*** 0,99 0,99 0,99 1 0,99 1 0,999 0,98 0,68*** 1,01 1 1 1 1 1 112 0,97 0,75*** 1,01 1,01 1 1 1,01 1 1,01
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd,PPI, PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFEof model 10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statisticalsignificance, respectively, for the Giacomini and White (2006) test with quadratic loss function. Number offorecasts is S = 300. The number of factors in the methods involving factor models is 1. The 1-step aheadforecasts range from February 1986 to January 2011. The 12-step ahead forecasts range from January 1987to December 2011.
30 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Table 1.2. MSFE ratios for first half of forecast sample (1 factor).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 0,96* 0,92 1,05 1,14 1,05 1,14 0,99 0,92 13 0,99* 0,74*** 1 0,99 0,98 1,01* 0,99 1 1
CPI 6 1,01 0,63** 1 1 1 1 1 1 19 0,99 0,66*** 1 1,01 1 1 1,01 1 1,0112 1,01 0,62*** 1,01 1 1 1 1 1 1
1 0,84 0,47*** 0,9 0,92 0,9 0,92 1,09* 1,06 13 0,79** 0,32*** 0,74** 0,73** 0,77* 0,81 0,91 0,94 0,94
FFR 6 0,74 0,32*** 0,89* 0,88* 0,94* 0,94* 0,94 0,97 0,959 0,91 0,51** 1,02 1,02 0,98*** 0,98** 1,07 0,99*** 1,0712 0,77 0,39*** 0,93 0,93 0,99 0,99 0,91 0,99 0,93
1 0,92* 1 1,15** 1,3*** 1,15** 1,3*** 0,99 0,89*** 13 1,01** 0,93 0,98** 0,99** 0,98 1,01 0,98 1,01 0,98*
PCEd 6 1 0,74*** 0,96 0,96 1 1* 0,96 1 0,969 1 0,84 1,01 1,01 1 1 1,01 1 1,0112 1 0,75*** 1 1 1 1 1 1 0,99
1 0,99 0,85 1 1,2 1 1,2 1 0,9 13 0,99 0,7*** 0,92* 0,92 0,94 0,99 0,97 0,99* 0,97
PPI 6 1,01 0,79** 0,96 0,95 1 1 0,96 1 0,969 1 0,75*** 0,94 0,95 1 1 0,99 1 0,9912 0,97 0,7 0,99** 0,99** 1* 1 0,99 1 0,99*
1 1 0,91 0,96 0,97 0,96 0,97 1,01 1,01 13 1 0,98 0,98 0,98 0,99 0,99 1 1 1
PEI 6 0,99 0,94*** 0,99*** 0,99*** 0,99*** 0,99*** 1 1** 19 0,98 0,94 0,96 0,95 1 1 0,98 1 0,9912 1 0,96*** 0,99 1,02 1 1 1,04 1 1,03
1 0,95 0,97 0,91 0,92* 0,91 0,92* 1,02 0,85 13 0,98 1 1,01 1 1,03 0,99* 1 0,92 1
UR 6 1,01* 0,91 1 0,99 1,01 0,98* 1,02** 0,97* 1,01**9 1,06 0,93 1 1 1,01 1 1,01 1 112 1,01** 0,83 0,95*** 0,92** 1 1* 0,95*** 1** 0,94**
1 0,96 0,95 0,89 0,93 0,89 0,93 0,96 1,02 0,97**3 1,07* 0,79*** 1,04 1,02 1,01 1 1 0,99 0,97
IP 6 1,03 0,82* 1 1 1 0,99 1 1 0,96*9 0,98 0,8** 1,02** 1,01*** 1 1 1,02* 1 0,9812 1,05 0,78*** 0,99** 0,98*** 1 1 1,01 1 1
1 0,93 0,86*** 0,99 0,99 0,99 0,99 1 0,93 13 1 0,91 0,99 0,99 1 0,99 0,99 1 0,99
RDI 6 0,99 0,82*** 1 1 1 1 1 1 1,019 1 0,92* 1 1 1* 1** 1 1* 1**12 0,99 0,88*** 1 1 1 1 1 1 1
1 0,67 1,15 1,32 1,72*** 1,32 1,72*** 0,99* 0,59* 13 1,08 0,97 1,03 1,03 0,96 1,07 1,03 1,08 1,03
PCE 6 1 0,85*** 0,99 0,98 1* 1 0,99 1 0,999 0,99 0,83** 1,02 1** 1 1 1** 1 112 0,99 0,84*** 1,02 1,03 1* 1 1,02 1* 1,02
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecastsis S′ = 150 (the first half of the S = 300 out-of-sample forecasts). The number of factors in the methodsinvolving factor models is 1. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-stepahead forecasts range from January 1987 to June 1999.
1.10. APPENDIX 31
Table 1.3. MSFE ratios for second half of forecast sample (1 factor).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 0,99 1,07 1,13 1,4 1,13 1,4 0,99 0,95 0,993 0,98 0,95 0,99 0,99 0,96 1,01 0,98 0,98 0,99
CPI 6 0,98 0,89* 0,99 0,94 1 1 0,99 1 0,999 0,98 0,83 1,01 1,01 1 1 0,98*** 1 0,98**12 1,02 0,91*** 1,01 1,02 1 1 1,03 1 1,03
1 0,72** 0,47*** 1,03 1,14 1,03 1,14 1,14 0,99 0,97***3 0,69*** 0,32*** 0,62** 0,57** 0,69* 0,77 0,72** 0,94 0,77**
FFR 6 0,69*** 0,21*** 0,9 0,85** 0,94 0,9 0,71** 0,97 0,74**9 0,81 0,27* 1,01 1,05 0,97 0,98 0,9 1 0,9212 0,73*** 0,36*** 0,89 0,91 1 1 0,83** 1,01* 0,84**
1 0,95 0,99 1,15 1,31 1,15 1,31 0,99* 0,92 0,993 0,99 0,9 1 1 0,98 0,99 0,99 0,97 0,99
PCEd 6 0,99 0,87* 1,01 1,01 0,98* 1 1 1 19 0,98* 0,88 1,01 1 1 1 1 1 112 1,01 0,91 0,95 0,95** 1* 1 0,97 1 0,97*
1 0,9** 0,89 1,13 1,29* 1,13 1,29* 0,99 0,76*** 0,993 0,98 0,89* 0,98 0,98 0,99 1 0,99 0,98 0,98
PPI 6 0,98 0,82 1,01 1,01 1,01 1 1 1 19 0,99 0,8 0,97 0,98 1 1 0,98 1 0,9712 1,06 0,95 1,06 1,04 1 1 1 1 1
1 1,08 1,01 0,91 0,9 0,91 0,9 1,08 1,06 1,013 1,04 0,85 0,92* 0,92* 0,94* 0,94* 1,08 1,03 1,05
PEI 6 1,04 0,82 0,98** 0,98** 0,98 0,98 1,05 1,01 1,049 1,03 0,87** 0,98 0,98 0,99 0,99 1,05* 1 1,04**12 0,99 0,72 0,93 0,92 1 1 0,91 1 0,91
1 1 1,07 0,88 0,79* 0,88 0,79* 1,11 0,53*** 13 1,1 1,02 0,98 0,9*** 0,95** 0,87** 1,1 0,68** 1,11
UR 6 1,07 0,78 0,94 0,99 0,98 0,93** 1,07 0,88** 1,099 1,01 0,86 0,92 0,99 0,98** 0,97* 0,98 0,95* 1,0212 0,95* 0,81 0,97 0,97 0,99 0,99* 0,94 0,99* 0,95
1 0,9* 1,01 0,9 0,84 0,9 0,84 0,98 0,9 13 1,04 1,13 1,11 1,09 1,09 0,96 1,06 0,95 1,05
IP 6 1,01 1 1,02 1,03 1,01 0,97 0,97 0,99 19 0,95 0,94 0,98* 0,97* 0,98 0,99 0,92*** 0,99 0,94**12 0,94* 0,85 0,97* 0,97 0,98 1 0,93 1 0,93
1 0,98 0,75** 1 1,03 1 1,03 1,01 0,96 13 0,99 0,76** 1 1 0,97 0,99 1,01 1 1,01*
RDI 6 0,99 0,75 0,99 0,99 0,99 1 1 1,01 19 0,98 0,83*** 1,01 1,01 1 1 1,02 1 1,0212 1 0,83*** 1 0,99 1 1 0,99 1 1
1 0,56** 0,85 1,48** 1,71** 1,48** 1,71** 1 0,56* 1,013 1,01 0,69** 1,05 1,05 1,03 1,04 1,04 1 1,04
PCE 6 0,97 0,63*** 0,99 0,99 0,99 0,99 0,99 0,99 0,999 0,97 0,53*** 1 1 1* 1 1 1 112 0,95 0,67*** 1 1 1 1 1 1 0,99
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (the second half of the S = 300 out-of-sample forecasts). The number of factors in the methodsinvolving factor models is 1. The 1-step ahead forecasts range from August 1998 to January 2011. The12-step ahead forecasts range from July 1999 to December 2011.
32 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Table 1.4. MSFE ratios for whole forecast sample (2 factors).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 0,98 1,04 1,12 1,34* 1,12 1,34* 0,99 0,99 13 0,98 0,9 0,99* 0,99* 0,96 1 0,97 1 0,98*
CPI 6 0,99 0,82** 0,99 0,95 1 1 0,98 1 0,999 0,98 0,79** 1,01 1,01 1 1 0,98 1 0,9712 1,02 0,83*** 1,01 1,01 1 1 1,03 1 1,02
1 1,37*** 0,83 1,66*** 1,75*** 1,66*** 1,75*** 1,88 1,68*** 13 0,78*** 0,33*** 0,71*** 0,67*** 0,76*** 0,82** 0,81 0,91 0,84***
FFR 6 0,74*** 0,27*** 0,92* 0,89*** 0,97* 0,95** 0,83 0,97* 0,84***9 0,87 0,37*** 1,03 1,04 0,98*** 0,99*** 0,96 0,99*** 0,9312 0,75*** 0,37*** 0,91** 0,92* 1 1 0,85 1 0,86***
1 0,94 0,99 1,15 1,31* 1,15 1,31* 1 0,95 13 1 0,91 0,99 1 0,98 1 0,99 1 0,98
PCEd 6 0,99 0,83*** 0,99 0,99 0,99* 1 0,99 1 19 0,99 0,87 1,01 1 1 1 1 1 0,9912 1 0,85*** 0,97 0,97** 1 1 0,98 1 0,97
1 0,92* 0,89 1,11 1,28** 1,11 1,28** 1,01 0,84*** 0,993 0,98 0,85*** 0,98 0,98 0,98 1 0,97 1 0,99
PPI 6 0,99 0,82 1 1 1,01 1 0,99 1 0,999 0,99 0,79 0,97 0,97 1 1 0,98 1 0,9812 1,04 0,9* 1,05 1,03 1 1 1,02 1 1,02
1 1 0,92 0,91 0,91 0,91 0,91 1,01 0,96 13 1,02 0,93 0,96* 0,96** 0,97 0,97 1,02 1,03 1,02
PEI 6 1,02 0,89 0,99*** 0,99*** 0,99* 0,99* 1,01 1,01 1,019 1 0,91** 0,97 0,97 1 1 1,01 1 1,0112 1 0,84 0,97 0,98 1 1 0,97 1 0,98
1 0,98 1,02 0,9 0,85* 0,9 0,85* 1,04 0,56*** 0,993 1,06 1,03 1,02 0,97** 1,01 0,94* 1,09 0,75*** 1,07
UR 6 1,06 0,84 0,98 1,01 1,01 0,97** 1,05 0,93** 1,069 1,04 0,9 0,96 1,01 1 0,99** 1,02 0,98** 0,9912 0,98*** 0,82 0,96** 0,95** 1 1** 0,95 0,99** 0,95**
1 0,94 1,01 0,92 0,89 0,92 0,89 0,99 1 0,95***3 1,05 0,98 1,08 1,06 1,06 0,97 1,04 0,98 1,01
IP 6 1,04 0,95 1,03 1,04 1,02 1 1,01 1,01 0,989 0,99 0,93* 1,03 1,02 1,03 1,03 0,98 1,03 112 1,01 0,86** 1,01 1,02 1,03 1,04 0,99 1,04 0,99
1 0,95 0,79*** 0,99 1,01 0,99 1,01 1 0,97 13 0,99 0,82*** 0,99 0,99 0,98 0,99 1 0,99 0,98
RDI 6 0,99 0,78** 0,99 0,99 1 1 0,98 1 0,999 0,99 0,87*** 1 1 1* 1 1 1 112 1 0,85*** 1 1 1 1 0,99 1 1
1 0,63*** 1,02 1,39** 1,74*** 1,39** 1,74*** 1 0,63** 13 1,05 0,83** 1,03 1,03 0,98 1,05 1,03 1,05 1,02
PCE 6 0,99 0,75*** 0,99 0,99 0,99* 1 0,99 1 0,979 0,98 0,68*** 1,01 1 1 1 0,98 1 112 0,97 0,75*** 1,01 1,02 1 1 1,01 1 0,99
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd,PPI, PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFEof model 10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statisticalsignificance, respectively, for the Giacomini and White (2006) test with quadratic loss function. Number offorecasts is S = 300. The number of factors in the methods involving factor models is 2. The 1-step aheadforecasts range from February 1986 to January 2011. The 12-step ahead forecasts range from January 1987to December 2011.
1.10. APPENDIX 33
Table 1.5. MSFE ratios for first half of forecast sample (2 factors).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 0,94 0,9 1,03 1,12 1,03 1,12 0,97 0,94 13 0,99* 0,74*** 1 0,98 0,98 1 0,99** 1,01 0,98
CPI 6 1,01 0,63** 1 1 1 1 1 1 0,999 0,99 0,66*** 1 1,01 1 1 1 1 0,9812 1,01 0,62*** 1,01 1 1 1 1 1 1,01
1 1,34** 0,75* 1,43** 1,47*** 1,43** 1,47*** 1,75*** 1,59*** 1,013 0,81** 0,33*** 0,76** 0,74** 0,79** 0,83* 0,89 0,91 0,93
FFR 6 0,76 0,33*** 0,91 0,9* 0,96* 0,96* 0,92 0,98* 0,939 0,92 0,51** 1,03 1,03 0,99*** 0,99** 1,08 0,99*** 1,0112 0,77 0,39*** 0,94 0,93 0,99 0,99 0,87 0,99 0,91
1 0,92* 1 1,14** 1,29*** 1,14** 1,29*** 0,99 0,9*** 13 1,01** 0,93 0,98** 0,99* 0,98 1,01 0,98 1,01** 0,98
PCEd 6 1 0,74*** 0,96 0,96 1 1* 0,96 1 0,969 1 0,84 1,01 1,01 1 1 1,01 1 1,0112 1 0,75*** 1 1 1 1 1 1** 0,99
1 0,99 0,85* 0,99 1,19 0,99 1,19 1,01 0,94 0,993 1 0,71*** 0,93 0,93 0,96 1 0,98 1 0,97
PPI 6 1,01 0,79** 0,96 0,95 1 1 0,96 1 0,969 1 0,75*** 0,94 0,95 1 1 0,99 1 0,9912 0,97 0,7 0,99** 0,99** 1 1 0,99 1 0,94
1 1 0,91 0,95 0,96 0,95 0,96 1 0,98 13 1 0,99 0,99 0,99 1 1 1,01 1 0,99
PEI 6 0,99 0,94*** 0,99*** 0,99*** 1*** 1*** 0,98 1* 19 0,98 0,94 0,96 0,95 1 1 0,99 1 0,9812 1 0,96*** 0,99 1,02 1 1 1 1 1,03
1 0,97 0,99 0,93 0,94 0,93 0,94 1,03 0,78* 13 0,98 1 1,02 1 1,04 0,99* 1 0,9* 1,01
UR 6 1,02* 0,91 1,01 0,99 1,01 0,98** 1,01** 0,97** 1,01*9 1,06 0,93 1,01 1,01 1,01 1 1,09 1 0,9512 1,01** 0,83 0,95*** 0,92** 1 1 0,98*** 1* 0,94***
1 0,95 0,94 0,88 0,92 0,88 0,92 0,95 1,03 0,96**3 1,08* 0,8*** 1,05 1,03 1,02 1,02 1,06 1,02 0,98
IP 6 1,03 0,82* 1 1 1 0,99* 1,01 1 0,979 0,97 0,8** 1,02* 1,01** 1 1 1,01 1 0,9812 1,05 0,78*** 0,99* 0,98** 1 1 1,05 1 1,02
1 0,93 0,86*** 0,99 0,99 0,99 0,99 0,99 0,94 13 1 0,91 0,99 0,99 1 0,99 0,99 1 0,96
RDI 6 0,99 0,82*** 1 1 1 1 1 1 19 1 0,92* 1 1 1* 1** 1 1* 1,0112 0,99 0,88*** 1 1 1 1 0,99 1 1,01
1 0,68 1,17 1,34 1,75*** 1,34 1,75*** 1,01** 0,65 0,99**3 1,07 0,97 1,02 1,03 0,96 1,06 1,02 1,07 1,03
PCE 6 1,01 0,85*** 0,99 0,98 1* 1 0,99 1 0,999 0,99 0,83** 1,02 1** 1 1 0,99*** 1 112 0,99 0,84*** 1,02 1,03 1 1 1,01 1 1
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (first half of the S = 300 out-of-sample forecasts). The number of factors in the methods involvingfactor models is 2. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-step aheadforecasts range from January 1987 to June 1999.
34 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Table 1.6. MSFE ratios for second half of forecast sample (2 factors).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 1 1,08 1,14 1,41 1,14 1,41 1 1,01 0,993 0,97 0,95 0,99 0,99 0,96 1 0,96 1 0,98
CPI 6 0,98 0,89* 0,99 0,94 1 1 0,98 1 0,999 0,98 0,83 1,01 1,01 1 1 0,98* 1 0,97***12 1,02 0,91*** 1,01 1,02 1* 1 1,03 1 1,02
1 1,4** 0,92 2,01*** 2,23*** 2,01*** 2,23*** 2,06*** 1,8*** 0,99**3 0,73*** 0,33*** 0,65** 0,6** 0,73* 0,82 0,73** 0,92 0,76***
FFR 6 0,71** 0,22*** 0,93 0,88** 0,97 0,93 0,73** 0,97 0,74***9 0,82 0,27* 1,02 1,06 0,98* 0,98* 0,84 1 0,8512 0,73*** 0,36*** 0,89 0,91 1 1* 0,82*** 1,01 0,82***
1 0,95 0,99 1,15 1,31 1,15 1,31 1 0,97 13 1 0,9 1 1 0,98 1 0,99 1 0,98
PCEd 6 0,99 0,87* 1,01 1,01 0,98* 1 1,01 1 1,019 0,98* 0,88 1,01 1 1 1 0,99 1 0,99*12 1,01 0,91 0,95 0,95** 1* 1** 0,98 1 0,97
1 0,91* 0,9 1,15 1,31* 1,15 1,31* 1,01 0,81** 0,993 0,98 0,89* 0,98 0,99 0,99 1 0,97 1 0,99
PPI 6 0,98 0,82 1,01 1,01 1,01 1 1 1 19 0,99 0,8 0,97 0,98 1 1 0,98 1 0,9712 1,06 0,95 1,06 1,04 1 1 1,03 1 1,04
1 1 0,94 0,84 0,83 0,84 0,83 1,03 0,95 13 1,04 0,85 0,92* 0,92* 0,94* 0,94* 1,05 1,06 1,07
PEI 6 1,05 0,83 0,99** 0,99** 0,99 0,99 1,05 1,02 1,039 1,03 0,87* 0,98 0,98 1 1 1,04 1,01 1,05**12 0,99 0,72 0,93 0,92 1 1 0,92 1 0,93
1 0,99 1,06 0,87 0,78* 0,87 0,78* 1,06 0,43*** 0,993 1,15 1,06 1,02 0,94*** 0,99 0,91** 1,17 0,67** 1,12
UR 6 1,1 0,8 0,96 1,02 1 0,95** 1,08 0,9** 1,19 1,02 0,87 0,93 1,01 0,99 0,98* 0,97 0,96** 1,0312 0,96 0,82 0,97 0,97 0,99 1* 0,93** 0,99* 0,96
1 0,94 1,06 0,94 0,87 0,94 0,87 1,01** 0,98 0,94***3 1,03 1,12 1,1 1,08 1,08 0,95 1,03 0,96 1,03
IP 6 1,04 1,03* 1,05 1,06 1,04 1 1,01 1,02 0,999 1 1* 1,03 1,02 1,04 1,05 0,97 1,05 112 1 0,9 1,03 1,03 1,04 1,06 0,97 1,06 0,97
1 0,98 0,74** 0,99 1,02 0,99 1,02 1,01 0,99 13 0,99 0,76** 0,99 0,99 0,97 0,99 1 0,99 1,01
RDI 6 0,99 0,75 0,99 0,99 0,99 1 0,97 1 0,999 0,98 0,83*** 1,01 1,01 1* 1 1,01 1 0,9912 1 0,83** 1 1 1 1 1 1 0,99
1 0,56** 0,85 1,48** 1,71* 1,48** 1,71* 1 0,61 13 1 0,68** 1,04 1,04 1,03 1,04 1,04 1,04 1,01
PCE 6 0,96 0,63*** 0,99 0,99 0,98 0,99 0,99 0,99 0,959 0,97 0,53*** 1 1 1 1 0,97 1 0,9912 0,95 0,67*** 1 1 1 1 1,01 1 0,99
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecastsis S′ = 150 (second half of the S = 300 out-of-sample forecasts). The number of factors in the methodsinvolving factor models is 2. The 1-step ahead forecasts range from August 1998 to January 2011. The12-step ahead forecasts range from July 1999 to December 2011.
1.10. APPENDIX 35
Table 1.7. MSFE ratios for whole forecast sample (3 factors).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 1 1,06 1,14 1,37** 1,14 1,37** 1 1,02 1,013 0,98 0,9 0,99 0,99 0,96 1 0,97 1 0,97
CPI 6 0,99 0,83** 1 0,95 1 1 0,99 1 0,999 0,98 0,79** 1,01 1,01 1 1 0,96 1 0,9712 1,02 0,83*** 1,01 1,01 1 1 1,01 1 1,02
1 1,15 0,69** 1,39*** 1,47*** 1,39*** 1,47*** 1,32 1,61*** 0,98**3 0,79** 0,34*** 0,72** 0,69** 0,78 0,84 0,83 1,06 0,94
FFR 6 0,71** 0,26*** 0,89 0,86 0,93 0,91 0,72 1,02 0,8***9 0,85 0,36*** 1 1,02 0,96 0,96 0,92 1,02 0,92*12 0,76** 0,38*** 0,92 0,94 1,01 1,01 0,78 1,03 0,9
1 0,93 0,98 1,13 1,29* 1,13 1,29* 1 0,94 13 1 0,91 0,99 0,99 0,98 1 0,98 1 0,98
PCEd 6 0,99 0,83*** 0,99 0,99 0,99 1 1 1 1*9 0,99 0,87 1,01 1 1 1 0,99 1 112 1 0,85*** 0,97 0,97** 1* 1 0,99 1 0,98
1 0,93* 0,9 1,13 1,3** 1,13 1,3** 1,01 0,86** 13 0,98 0,85*** 0,98 0,98 0,98 1 0,97 1 0,98
PPI 6 0,99 0,82 1 1 1,01 1 0,99 1 0,999 0,99 0,79 0,97 0,97 1 1 0,98 1 0,9712 1,04 0,9* 1,05 1,03 1 1 1,06 1 1,03
1 0,97 0,9** 0,88* 0,88* 0,88* 0,88* 0,99 0,97 13 0,99 0,9*** 0,93** 0,93** 0,95* 0,95* 1 1,02 0,99
PEI 6 0,99 0,87* 0,96** 0,96* 0,97** 0,97** 0,98 1,01 0,99*9 0,99 0,9*** 0,96 0,96 0,99 0,99 0,99 1* 0,9912 0,99 0,84 0,97 0,98 1 1 0,97 0,99 0,97
1 0,89*** 0,93 0,82*** 0,78*** 0,82*** 0,78*** 0,94 0,49*** 1,013 0,98 0,95 0,94** 0,89** 0,93 0,87** 0,99 0,6*** 0,99
UR 6 0,99 0,78 0,91** 0,94 0,94 0,9* 0,99 0,76** 0,98**9 0,99 0,86 0,92* 0,96 0,95 0,94 1 0,85 0,91**12 0,97*** 0,82 0,95* 0,94** 0,99 0,99 0,97 0,94* 0,94**
1 0,92*** 0,99 0,9 0,87 0,9 0,87 0,96 1 0,97**3 1,06 0,99 1,09 1,07 1,07 0,98 1,07 1,01 1,02
IP 6 1,03 0,94 1,02 1,03 1,01 0,99 1,03 1,01 0,979 0,98 0,92 1,02 1,01 1,02 1,02 0,99 1,01 0,9812 1 0,85** 1 1 1,02 1,03 0,99 1,01 0,96
1 0,94 0,78*** 0,98 1 0,98 1 0,99 0,96 13 0,99 0,82*** 0,99 0,99 0,98 0,99* 0,99 0,99 0,98
RDI 6 0,98 0,77** 0,99 0,99 0,99 0,99 0,97 0,99 0,98*9 0,98 0,87*** 1 1 1 1 0,98 0,99 0,9912 1 0,85*** 1 1 1 1 0,99 1 0,99
1 0,64*** 1,03 1,4*** 1,75*** 1,4*** 1,75*** 1 0,65** 0,99*3 1,04 0,83** 1,03 1,03 0,98 1,05 1,02 1,05 1,01
PCE 6 0,99 0,75*** 0,99 0,99 0,99 1 0,97 1 0,979 0,98 0,68*** 1,01 1 1 1 0,98 1 0,9912 0,97 0,75*** 1,01 1,01 1 1 0,98 1 0,98
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd,PPI, PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFEof model 10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statisticalsignificance, respectively, for the Giacomini and White (2006) test with quadratic loss function. Number offorecasts is S = 300. The number of factors in the methods involving factor models is 3. The 1-step aheadforecasts range from February 1986 to January 2011. The 12-step ahead forecasts range from January 1987to December 2011.
36 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Table 1.8. MSFE ratios for first half of forecast sample (3 factors).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 0,93 0,89 1,02 1,1 1,02 1,1 0,95* 0,92 0,993 0,99* 0,74*** 1 0,98 0,98 1 0,99** 1 0,98
CPI 6 1,01 0,63** 1* 1* 1 1 1,02 1 0,999 0,99 0,66*** 1 1,01 1 1 1 1 0,9912 1,01 0,62*** 1,01 1 1 1 1 1 1
1 1,15 0,65*** 1,23 1,27 1,23 1,27 1,24** 1,49*** 0,983 0,8 0,33*** 0,75* 0,73* 0,78 0,82 0,85 0,98 1
FFR 6 0,68* 0,3*** 0,82 0,81 0,87 0,87 0,69* 0,94 0,869 0,84 0,47** 0,94 0,94 0,9* 0,9* 0,92 0,94 0,9612 0,73* 0,37*** 0,88 0,88 0,94 0,94 0,75 0,96 0,88
1 0,9** 0,98 1,12** 1,27*** 1,12** 1,27*** 0,96* 0,88*** 0,97**3 1,01** 0,93 0,98* 0,99* 0,98 1,01 0,97 1,01** 0,98
PCEd 6 1 0,74*** 0,96 0,96 1 1 0,97 1* 0,969 1 0,84 1,01 1,01 1 1 1,01 1 1,0112 1 0,75*** 1 1 1 1 1 1 0,98
1 1 0,86 1 1,2 1 1,2 1,02 0,95 0,99**3 1,01 0,71*** 0,94 0,94 0,96 1,01 0,97 1,01 0,96
PPI 6 1,01 0,79** 0,96 0,95 1 1 0,98 1 0,979 1 0,75*** 0,94 0,95 1 1 0,99 1 0,9912 0,97 0,7 0,99** 0,99** 1 1 0,97* 1 0,93
1 0,99 0,9 0,94 0,95 0,94 0,95 1 0,98 13 1 0,98 0,98* 0,98 0,99 0,99 1,01 1 1
PEI 6 0,98 0,93*** 0,98*** 0,98*** 0,99** 0,99** 0,97* 1,01** 19 0,98 0,94 0,96 0,95 1 1 0,96 1 0,9812 0,99 0,96*** 0,99 1,02 1 1 1 1 1,02
1 0,91 0,93 0,87 0,88* 0,87 0,88* 0,97 0,7** 1,013 0,97 0,99 1,01 0,99 1,03* 0,98** 0,98 0,83** 0,98
UR 6 1 0,9 0,99 0,97 1 0,97* 1* 0,92** 0,999 1,04 0,92 0,99 0,99 1 0,99 1,04 0,97* 0,9***12 1** 0,83 0,94*** 0,92*** 0,99 0,99** 0,97*** 0,98** 0,93***
1 0,87* 0,86 0,81** 0,84* 0,81** 0,84* 0,89* 0,99 0,973 1,09 0,8** 1,06 1,04 1,03 1,02 1,05 1,02 0,98
IP 6 1,03 0,82** 1 1 1 0,99 1,01 1,01 0,979 0,97 0,8** 1,01 1,01 1 1 0,97 1 0,9712 1,05 0,78*** 0,99 0,98* 1 1 1,04 1 0,99
1 0,92 0,85*** 0,98 0,98 0,98 0,98 0,98 0,93 13 1 0,9 0,99 0,99 0,99 0,99 0,99 0,99 0,95
RDI 6 0,99 0,81*** 0,99 0,99 0,99** 0,99** 0,99 0,99** 19 0,99 0,92* 0,99 0,99 1* 1* 0,99 1* 1,0212 0,99 0,88*** 1 1 1 1 0,99 1 1
1 0,68 1,17 1,35 1,76*** 1,35 1,76*** 1,01** 0,67 13 1,07 0,97 1,02 1,03 0,96 1,06 1,03 1,06 1,02
PCE 6 1,01 0,85*** 0,99 0,98 1* 1 0,99 1 0,989 0,99 0,83** 1,02 1** 1 1 0,99 1 1,0112 0,99 0,84*** 1,02 1,03 1** 1* 0,99 1* 0,99
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (first half of S = 300 out-of-sample forecasts). The number of factors in the methods involvingfactor models is 3. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-step aheadforecasts range from January 1987 to June 1999.
1.10. APPENDIX 37
Table 1.9. MSFE ratios for second half of forecast sample (3 factors).
h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9
1 1,02 1,11 1,17 1,45** 1,17 1,45** 1,01 1,04 1,013 0,97 0,95 0,99 0,99 0,96 1 0,96 1 0,97
CPI 6 0,98 0,89* 0,99 0,94 1 1 0,99 1 0,999 0,98 0,83 1,02 1,01 1 1 0,95 1 0,97***12 1,02 0,91*** 1,01 1,02 1 1 1,01 1 1,02
1 1,15 0,75 1,64** 1,82*** 1,64** 1,82*** 1,41*** 1,77*** 0,983 0,78 0,36*** 0,7 0,65 0,78 0,87 0,81 1,17 0,89
FFR 6 0,74 0,23** 0,97 0,92 1,01 0,97 0,76 1,13 0,74**9 0,87 0,29* 1,08 1,12 1,03 1,04 0,91 1,11 0,89*12 0,8 0,39** 0,97 1 1,09 1,09 0,82 1,11 0,92
1 0,94 0,98 1,13 1,29 1,13 1,29 1,01 0,96 13 1 0,9 1 1 0,98 1 0,99 1 0,99
PCEd 6 0,99 0,87** 1,01 1,01 0,99 1 1,01 1 1,02*9 0,98* 0,88 1,01 1 1 1 0,98 1 0,9912 1,01 0,91 0,95 0,95** 1** 1 0,99 1 0,98
1 0,92* 0,91 1,16 1,33* 1,16 1,33* 1 0,84** 13 0,98 0,89* 0,98 0,99 0,99 1 0,97 1 0,99
PPI 6 0,99 0,83 1,01 1,02 1,01 1 1 1 0,999 0,99 0,8 0,97 0,98 1 1 0,98 1 0,9712 1,06 0,95 1,06 1,04 1 1 1,08 1 1,05
1 0,95 0,9 0,8 0,79 0,8 0,79 0,98 0,96 1,013 0,99 0,81** 0,87* 0,87* 0,89* 0,89* 0,99 1,05 0,99
PEI 6 1 0,79 0,94 0,94 0,94 0,94 0,99 1,02 0,979 1,01 0,85** 0,96 0,96 0,98 0,98 1,02 1,01* 1,0112 0,99 0,73 0,93 0,92 1 1 0,94 0,99 0,91
1 0,86** 0,93 0,77** 0,68*** 0,77** 0,68*** 0,9** 0,36*** 1,013 0,98 0,91 0,87*** 0,81** 0,85*** 0,78** 0,99 0,47** 0,99
UR 6 0,98 0,71 0,85** 0,91 0,89 0,85* 0,99** 0,66** 0,97***9 0,95 0,81 0,86* 0,93 0,92 0,91 0,96 0,78 0,9212 0,95** 0,81 0,96 0,96 0,98 0,99 0,97 0,91* 0,94
1 0,94 1,06 0,95 0,88 0,95 0,88 0,99 1 0,973 1,04 1,13 1,1 1,09 1,09 0,95 1,08 1,01 1,04
IP 6 1,03 1,01 1,03 1,05 1,02 0,99 1,04 1,01 0,979 0,99 0,98 1,02 1,01 1,03 1,03 0,99 1,02 0,9812 0,98 0,88 1,01 1,01 1,02 1,04 0,97 1,01 0,95
1 0,96 0,73*** 0,98 1,01 0,98 1,01 0,99 0,98 13 0,99 0,76*** 0,99 0,99 0,97 0,99* 0,99 0,98 1
RDI 6 0,98 0,74 0,98 0,98 0,98 0,99 0,95 0,98 0,979 0,97 0,83*** 1 1 0,99 0,99 0,98 0,99 0,9712 1 0,83*** 1 0,99 1 1 0,99 1 0,98
1 0,57** 0,86 1,5** 1,73* 1,5** 1,73* 0,99 0,62 0,99*3 1 0,68** 1,04 1,04 1,03 1,04 1,01 1,04 1
PCE 6 0,96 0,63*** 0,99 0,99 0,98 0,99 0,95 0,99 0,949 0,97 0,53*** 1 1 1 1 0,96 1 0,9712 0,95 0,67*** 1 1 1 1 0,97 1* 0,96
Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (second half of S = 300 out-of-sample forecasts. The number of factors in the methods involvingfactor models is 3. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-step aheadforecasts range from January 1987 to June 1999. The 1-step ahead forecasts range from August 1998 toJanuary 2011. The 12-step ahead forecasts range from July 1999 to December 2011.
38 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Table
1.10.Varian
ceratio
sr
j,nt
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CP
IF
FR
PC
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PP
IP
EI
UR
IPR
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PC
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1facto
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r1,346E
-070,0063
4,26E-08
3,75E-07
1,38E-07
1,71E-02
2,61E-05
4,16E-07
1,11E-06
2facto
rs1
stfacto
r8,23E
-060,0068
2,28E-06
1,30E-05
5,34E-07
2,89E-03
7,09E-06
3,24E-07
1,04E-07
2n
dfacto
r6,93E
-050,02
2,33E-05
0,000158,07E
-062,49E
-037,39E
-066,83E
-061,08E
-05
3facto
rs1
stfacto
r5,63E
-060,04
1,28E-06
8,42E-06
3,93E-07
3,30E-03
8,09E-05
4,72E-08
5,55E-07
2n
dfacto
r0,00017
0,00975,37E
-050,00017
5,82E-06
2,56E-03
3,27E-03
5,53E-06
4,17E-06
3rd
factor
8,24E-06
0,093,56E
-064,14E
-062,17E
-072,68E
-038,80E
-041,10E
-061,10E
-06
Valu
esfo
raverage
variance
ratior
j,nt
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the
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samp
le.
1.10. APPENDIX 39
Table 1.11. MSFE ratios, cosine basis loadings.
ρ f2,y = 0.5 ρ f1,y = 0.5
ρ f1,y h mod 8 mod 9 ρ f2,y h mod 8 mod 9
1 1,01 1,02 1 0,98 0,943 0,99 1 3 0,97 1,01
0.1 6 1 1,01 0.1 6 1 1,019 1,01 1 9 0,96** 112 0,98 1 12 0,97 0,99
1 0,95** 0,73*** 1 0,98 0,64***3 1 0,99 3 0,99 0,95
0.2 6 0,98 1,01* 0.2 6 0,96*** 19 1 1 9 0,97 112 0,99 1 12 0,98 1
1 0,98 0,91* 1 0,96 13 0,98** 1 3 0,98 0,98
0.3 6 0,98 1 0.3 6 0,98 0,999 1 1 9 0,97*** 112 0,97** 1 12 0,98** 1
1 0,98 0,6*** 1 0,92** 0,85***3 0,98 0,96 3 0,99 0,98
0.4 6 0,97 1,01 0.4 6 0,98 0,999 0,96*** 1 9 0,98 112 0,98 1 12 0,98 1
1 1,01 0,99 1 0,94** 0,34***3 0,99* 1 3 0,93** 0,74**
0.5 6 0,98 1 0.5 6 0,94 0,979 0,98** 1 9 0,98 1,0312 0,99 1 12 1 1,03
1 0,97 0,75*** 1 0,92** 0,87***3 0,98 0,99 3 1 1,01
0.6 6 1 1,02 0.6 6 0,99 19 1,01 1,02 9 0,98 112 0,97** 1,02 12 0,99 1
1 0,92** 0,86*** 1 0,95** 0,993 0,97 1,01 3 0,97 1,01
0.7 6 0,98 1 0.7 6 0,99 19 0,99 1 9 0,99 1,0112 0,99 1 12 0,98** 1,01
1 0,96 1,03 1 0,98 0,86***3 1,01 1,03* 3 0,96 0,99
0.8 6 0,99 1 0.8 6 0,99 19 1,01 1 9 0,95 112 0,99** 1 12 0,98 1
1 0,96 0,81*** 1 1,03 1,033 0,98 0,97 3 1,02 1,01
0.9 6 0,98 1,03 0.9 6 1** 1,019 0,99 1,02 9 1 1,01*12 1 1 12 0,99 1*
Simulated data with cosine basis loadings. MSFE ratios between model 10 and models 8 and 9 for differentlevels of correlation between factors and forecast objective. Forecasting leads h = 1,3,6,9,12. A value lowerthan one indicates a lower MSFE of model 10 w.r.t. the competing models. One, two, and three stars mean.10, .05, and .01 statistical significance, respectively, for the Giacomini and White (2006) test with quadraticloss function. Number of forecasts is S = 300. The number of latent factors is 2. The correlation betweenthe two latent factors is ρ f1 , f2
= 0.10.
40 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS
Table 1.12. MSFE ratios, random loadings.
ρ f2,y = 0.5 ρ f1,y = 0.5
ρ f1,y h mod 8 mod 9 ρ f2,y h mod 8 mod 9
1 1 0,44*** 1 1 0,963 0,98* 0,94 3 0,97 0,98
0.1 6 0,98 1,01* 0.1 6 0,98 19 0,97 1,01 9 0,96 112 0,98 1 12 0,96 1
1 1,04*** 0,9** 1 0,99 0,91***3 1,08** 1,1*** 3 0,97** 0,97
0.2 6 1,08* 1,08** 0.2 6 0,96* 0,989 1,04 1,07* 9 0,95 0,9812 1,03 1,05 12 0,96 0,98
1 0,93 0,9 1 1 0,85***3 0,97 1 3 0,99 1
0.3 6 0,98 1 0.3 6 1 19 0,98 1 9 0,98 112 0,97 1 12 0,98 1
1 1 0,28*** 1 0,99 0,89*3 0,99 0,66*** 3 0,97 0,98
0.4 6 0,97* 0,84* 0.4 6 0,97 0,999 0,97** 0,94 9 0,96* 0,9912 0,96 0,94 12 0,96* 0,99
1 0,98 0,34*** 1 0,97 0,5***3 0,98* 0,79*** 3 1 0,6***
0.5 6 0,97 0,99 0.5 6 1 0,78***9 1 1,02 9 0,99 0,88**12 1,01 1 12 0,98 0,95
1 0,97 0,87*** 1 1,01 0,32***3 1,01 1,01 3 0,99 0,72***
0.6 6 1,01 1,02 0.6 6 0,99 0,9**9 1,01 1,01 9 0,98 0,9612 0,99 0,99 12 0,98 1
1 1 0,65*** 1 1,01 0,77***3 0,99* 0,97* 3 0,98 0,96
0.7 6 0,99 1 0.7 6 0,99 0,989 0,99 1* 9 0,98 0,9712 0,98 1** 12 0,97 0,99
1 0,98 0,25*** 1 1,02 0,65***3 1,02* 0,77*** 3 1,04 0,86**
0.8 6 0,98 0,96 0.8 6 1,05 0,949 0,97 0,99 9 1,03 0,9912 0,97 0,99 12 1,03 1,04
1 0,99 0,95 1 1 0,29***3 0,99 1 3 1,01** 0,56***
0.9 6 1,01 1 0.9 6 0,99 0,839 0,98 1 9 0,97 0,9112 0,99** 1 12 0,99 0,98
Simulated data with random loadings. MSFE ratios between model 10 and models 8 and 9 for differentlevels of correlation between factors and forecast objective. Forecasting leads h = 1,3,6,9,12. A value lowerthan one indicates a lower MSFE of model 10 w.r.t. the competing models. One, two, and three stars mean.10, .05, and .01 statistical significance, respectively, for the Giacomini and White (2006) test with quadraticloss function. Number of forecasts is S = 300. The number of latent factors is 2. The correlation betweenthe two latent factors is ρ f1 , f2
= 0.10.
CH
AP
TE
R
2THE FORECASTING POWER OF THE YIELD CURVE
A SUPERVISED FACTOR MODEL APPROACH
Lorenzo Boldrini
Aarhus University and CREATES
Eric T. Hillebrand
Aarhus University and CREATES
41
42 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
Abstract
We study the forecast power of the yield curve for macroeconomic time series, such
as consumer price index, personal consumption expenditures, producer price in-
dex, real disposable income, unemployment rate, and industrial production. We
employ a state-space model in which the forecasting objective is included in the state
vector. This amounts to an augmented dynamic factor model in which the factors
(level, slope, and curvature of the yield curve) are supervised for the macroeconomic
forecast target. In other words, the factors are informed about the dynamics of the
forecast objective. The factor loadings have the Nelson and Siegel (1987) structure and
we consider one forecast target at a time. We compare the forecasting performance
of our specification to benchmark models such as principal components regression,
partial least squares, and ARMA(p,q) processes. We use the yield curve data from
Gürkaynak, Sack, and Wright (2006) and Diebold and Li (2006) and macroeconomic
data from FRED. We compare the models by means of the conditional predictive
ability test of Giacomini and White (2006). We find that the yield curve has more
forecast power for real variables compared to inflation measures and that supervising
the factor extraction for the forecast target can improve forecast performance.
2.1. INTRODUCTION 43
2.1 Introduction
The forecasting power of the yield curve for macroeconomic variables has been docu-
mented in many papers, see among others Harvey (1988), Stock and Watson (1989),
Estrella and Hardouvelis (1991), and Chinn and Kucko (2010). However, the predictive
power of the yield curve has changed through the years, see for instance Giacomini
and Rossi (2006), Rudebusch and Williams (2009), and Stock and Watson (1999a)
raising doubt about its reliability as a predictor.
The aim of the present paper is twofold. The first one is to analyse the forecast
power of the yield curve for macroeconomic variables. This is carried out by com-
paring forecasting models that make use of the yield curve information to ones that
do not. We also study the stability of the yield curve as a predictor, by considering
different time spans. The second objective is to assess the forecast performance
of a supervised factor model, as proposed in Boldrini and Hillebrand (2015). This
model is a particular specification of a factor model in which the factors are extracted
conditionally on the forecast target.
There exists an extensive literature on the forecasting power of the yield curve.
One of the first works testifying the forecasting power of the yield curve for macroe-
conomic variables was Harvey (1988), who within the framework of the consumption
based asset pricing model found that the real term structure of interest rates is a
good predictor for consumption growth. Within the framework of dynamic factor
models, Stock and Watson (1989) find that two interest rate spreads, namely the
difference between the six-month commercial paper and the six-month Treasury bill
rates, and the difference between the ten-year and one-year Treasury bond rates, are
good predictors of real activity. In related papers, Bernanke (1990) and Friedman and
Kuttner (1993), using linear regressions, find that the spread between the commercial
paper rate and the Treasury bill rate is a particularly good predictor for real activity
indicators and inflation. In Estrella and Hardouvelis (1991) the authors conclude
that a positive slope of the yield curve is associated with a future increase in real
economic activity. They find that it outperforms both in-sample and out-of-sample
other variables, such as the index of leading indicators, real short-term interest rates,
lagged growth in economic activity, and lagged rates of inflation as well as survey
forecasts. Still in the framework of linear regressions Kozicki (1997) and Hamilton and
Kim (2000) confirm the predictive power of the spread for real growth and inflation.
Ang, Piazzesi, and Wei (2006) build a dynamic model for GDP growth and yields that
does not allow arbitrage and completely characterizes expectations of GDP. Contrary
to previous findings, they find that the short rate has more predictive power than any
term spread.
Diebold and Li (2006) focus on forecasting the yield curve by means of the Nelson
and Siegel (1987) (NS) model. They interpret the dynamically moving parameters
as level, slope, and curvature. Diebold et al. (2006) cast the NS model in state-space
form and analyse the correlations between the extracted factors and macroeconomic
44 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
variables. They find a strong correlation between the level factor and inflation and
between the slope factor and capacity utilization.
Giacomini and Rossi (2006) examine the stability of the forecasting power of the
yield curve for economic growth in the US economy, using forecast breakdown tests.
They find a forecast breakdown during the periods 1974-76 and in 1979-87, the Burns-
Miller and the Volcker monetary regimes respectively, whereas during 1987-2006,
corresponding to when Alan Greenspan was chairman of the FED, the yield curve
proved to be a more reliable forecaster for real growth. Similarly, Stock and Watson
(1999a) found some evidence of structural breaks in the relationship between the
slope of the yield curve and real activity during the past years.
The ability to extract information from large datasets has made factor models
an appealing tool in forecasting. Stock and Watson (1999b) and Stock and Watson
(2002a), for instance, investigate forecasts of output growth and inflation using a large
number of economic indicators, including many interest rates and yield spreads. The
advantage of factor models is that the information contained in a (potentially) large
number of predictors can be summarized in a few factors. Comprehensive surveys
on factor models can be found in Bai and Ng (2008b), Breitung and Eickmeier (2006),
and Stock and Watson (2011).
In the standard approach to factor models, the extracted factors are the same for
all the forecast targets. One of the directions the literature has taken for improving
on this approach is to select factors based on their ability to forecast a specific target.
Different methods have been proposed in the literature that address this problem.
The method of partial least squares regression (PLSR), for instance, constructs a set
of linear combinations of the inputs (predictors and forecast target) for regression,
for more details see for instance Friedman et al. (2001). Bai and Ng (2008a) proposed
performing PCA on a subset of the original predictors, selected using thresholding
rules. This approach is close to the supervised PCA method proposed in Bair et al.
(2006), that aims at finding linear combinations of the predictors that have high
correlation with the target. In particular, first a subset of the predictors is selected,
based on the correlation with the target (i.e. the regression coefficient exceeds a given
threshold), then PCA is applied on the resulting subset of variables. Bai and Ng (2009)
consider ‘boosting’ (a procedure that performs subset variable selection and coeffi-
cient shrinkage) as a methodology for selecting the predictors in factor-augmented
autoregressions. Finally, Giovannelli and Proietti (2014) propose an operational su-
pervised method that selects factors based on their significance in the regression of
the forecast target on the predictors.
Hillebrand, Huang, Lee, and Li (2012) propose a method to exploit the yield curve
information in forecasting macroeconomic variables. The model is a modified NS
factor model, where the new NS yield curve factors are supervised for a specific vari-
able to forecast. They show that it outperforms the conventional (non-supervised)
NS factor model in out-of-sample forecasting of monthly US output growth and
2.2. DYNAMIC FACTOR MODELS AND SUPERVISION 45
inflation.
In this paper we assess the forecast performance of the yield curve, using a super-
vised factor model, as presented in Boldrini and Hillebrand (2015). In the supervised
framework, the factors are informed of the forecast target (supervised) and the model
has a linear, state-space representation to which Kalman filtering techniques apply.
In particular, we select the Nelson and Siegel (1987) factor structure for the yield
curve. The latent factors in this specification are three and represent the level, slope,
and curvature of the yield curve. We consider also time variation in the factor loadings
using a specification similar to the one used in Koopman, Mallee, and Van der Wel
(2010). We include one forecast target at a time in the state vector, together with the
three NS factors. This allows us to estimate the latent factors using information also
in the forecast target, through the Kalman filter recursions. The factor extraction is
thus conditional on the forecast target.
We compare the forecasting performance of the proposed specification to that of
models that make use of the yield curve information (principal components regres-
sion, partial least squares, two-step forecasting procedures as in Stock and Watson
(2002a)) and models that do not (AR(p) and M A(q) processes). We use a rolling
windows scheme and consider both direct and indirect h-step ahead forecasts. We
compare the forecast performance of the different models by means of the Giacomini
and White (2006) test, considering different time spans. We use yield curve data from
Gürkaynak et al. (2007) and Diebold et al. (2006), and macroeconomic variables from
FRED. All the data is relative to the US economy. The selected forecast objectives are
consumer price index (CPI), personal consumption expenditures (PCE), producer
price index (PPI), real disposable income (RDI), unemployment rate (UR), and indus-
trial production (IP).
The paper is organized as follows: in Section 2.2 we introduce the supervised
factor model and relate it with other forecasting methods based on factor models; in
Section 2.3 we provide some details on the computational aspects of the analysis; in
Sections 2.4 we describe the empirical application; finally, Section 2.5 concludes.
2.2 Dynamic factor models and supervision
Let xt be the forecast objective, yt = [y1t , . . . , y N
t ] an N -dimensional vector of predic-
tors, h the forecast lead, and T the last available observation in the estimation window.
Throughout we indicate with a “caret”, estimates of scalars, vectors, or matrices.
Supervised factor model
In this section we present the supervised factor model as in Boldrini and Hillebrand
(2015), that we use to assess the forecasting power of the yield curve. Consider the
46 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
system
[yt
xt
]=
[Λ 00 1
][ft
xt
]+
[εt
0
], εt ∼ N (0,H),[
ft+1
xt+1
]= c+T
[ft
xt
]+ηt , ηt ∼ N (0,Q), (2.1)
where ft ∈Rk are latent factors,Λ is a matrix of factor loadings, T and c are a matrix
and a vector of coefficients, respectively, of suitable dimensions, εt ∈ RN and ηt ∈Rk+1 are vectors of disturbances and H and Q are their respective variance-covariance
matrices.
In this supervised framework, the forecast objective is placed in the state equation
together with the latent factors and the predictors are modelled in the measurement
equation. The intuition behind the model is that if the forecast objective is correlated
with the factors then their estimation by means of the Kalman filter will benefit
from the inclusion of the forecast target in the state vector. This follows because for a
general linear state-space system, the Kalman filter delivers the best linear predictions
of the latent states at time t , given the information from all the observables entering
the measurement equation up to and including time t . In the particular case of
Gaussian innovations, as in the system (2.1), the best linear prediction coincides
with the conditional expectation. For more details on the optimality properties of the
Kalman filter see for instance Brockwell and Davis (2009).
The forecasting scheme for this model is the following:
(i) the system parameters are estimated by maximizing the likelihood function,
delivered by the Kalman filter recursions;
(ii) given the parameter estimates, the Kalman filter is run on the data;
(iii) indicating the state vector with αt = [f′t , xt ], the forecast xT+h is then
xT+h = [0′,1]
[T
hαT +
h−1∑i=0
Tic
],
where αT represents the filtered αT .
Note that the last element of the filtered state vector at time t corresponds to xt .
To be able to compute direct h-period ahead forecasts, the state-space form (2.1)
can be modified extending the state vector by including h −1 lags of the factors. The
system (2.1) thus becomes
2.2. DYNAMIC FACTOR MODELS AND SUPERVISION 47
[yt
xt
]=
[Λ(N )×(K−1) 0N×1 0N×K (h−1)
01×(K−1) 1 01×K (h−1)
]
ft
xt...
ft−(h−1)
xt−(h−1)
+
[εt
0
],
ft+1
xt+1...
ft−h
xt−h
=
[c f
0K×(h−1)
]+
[0K×(h−1) T f
I(h−1)×(h−1) 0(h−1)×K
]
ft
xt...
ft−(h−1)
xt−(h−1)
+
[IK×K
0(h−1)×K
]ηt ,
(2.2)
where T f is a matrix of parameters relating the state vector at times t and t +h, c f
is a vector of parameters, and ηt is the same as in equation (2.1). The state vector
can be recognized to be a V AR(1) representation of a restricted V AR(h −1). For the
representation of V AR(p) processes in state-space form see for instance Aoki (1990).
The forecasting scheme for the direct approach is analogous to the previous one.
Two-step procedure
Forecasting using dynamic factor models (DFM hereafter) is often carried out in a
two-step procedure as in Stock and Watson (2002a). Consider the model
xt+h = β(L)′ft +γ(L)xt +εt+h , (2.3)
yt ,i = λi (L)ft +ηt ,i , (2.4)
with i = 1, . . . , N and where ft = ( ft ,1, . . . , ft ,k ) are k latent factors, ηt = [ηt ,i , . . . ,ηt ,N ]′
and εt are idiosyncratic disturbances, β(L) = ∑qj=0β j+1L j , λi (L) = ∑p
j=0λi ( j+1)L j ,
and γ(L) = ∑sj=0γ j+1L j are finite lag polynomials in the lag operator L; β j ∈ Rk ,
γ j ∈ R, and λi j ∈ R are parameters and q, p, s ∈N0 are indices. The assumption on
the finiteness of the lag polynomials allows us to rewrite (2.3)-(2.4) as a static factor
model, i.e. a factor model in which the factors do not appear in lags:
xt+h = c +β′Ft +γ(L)xt +εt+h ,
yt = ΛFt +ηt , (2.5)
with Ft = [f′t , . . . , f′t−r ]′, r = max(q, p), the i -th row of Λ is [λi ,1, . . . ,λi ,r+1], and β =[β′
1, . . . ,β′r+1]′. The forecasting scheme is the following:
48 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
(i) extraction of the factors ft from the predictors xt modelled in equation (2.4)
using either principal components, as in Stock and Watson (2002b), or the
Kalman filter;
(ii) regression of the forecast objective on the lagged estimated factors and on its
lags according to the forecasting equation (2.3) with t = 1, ...,T −h;
(iii) the forecast is obtained from the estimated factors and regression coefficients
as
xT+h = c + β′Ft + γ(L)xT . (2.6)
Stock and Watson (2002a) developed theoretical results for this two-step procedure, in
the case of principal components estimation. In particular, they show the asymptotic
efficiency of the feasible forecasts and the consistency of the factor estimates.
In our empirical application we estimate this models using the NS factor loadings
and extract the factors using the Kalman filter, we then use an auxiliary equation to
compute the forecasts.
The difference between the supervised model (2.1) and model (2.5) is that in the
latter the extraction of the factors is independent of the forecast objective whereas in
the former one the extracted factors are informed (supervised) of the specific forecast
target considered.
Nelson-Siegel factor model
We consider here the forecast of macroeconomic variables using the term structure
of interest rates as predictor. In the 1990s, factor models have gained popularity in
modelling the yield curve, for instance with the works of Litterman and Scheinkman
(1991) and Knez, Litterman, and Scheinkman (1994), who used factor analysis to
extract common features from yield curves in different countries and periods. They
concluded that three factors explained the greater part of the variation in the yield
curve. Nelson and Siegel (1987) proposed a way to model the yield curve based on
three functions describing level, slope, and curvature, of the term structure of interest
rates.
Following Diebold et al. (2006) we cast the Nelson and Siegel (1987) term struc-
ture model in state-space form and extract three factors whose interpretation is that
of level, slope, and curvature (in the following they are labelled LVt , SLt , and CVt
respectively). The latent factors corresponding to level, slope, and curvature are unre-
stricted whereas the factor loadings are restricted to have the Nelson-Siegel structure.
This guarantees positive forward rates at all horizons and a discount factor that ap-
proaches zero as maturity increases. The supervised factor model (2.1) becomes
then
2.2. DYNAMIC FACTOR MODELS AND SUPERVISION 49
yτ1
t...
yτNt
xt
= Λ
N×3(λ) 0
N×1
01×3
1
LVt
SLt
CVt
xt
+
ετ1t...
ετNt
0
,
LVt+1
SLt+1
CVt+1
xt+1
= c+T
LVt
SLt
CVt
xt
+ηt , (2.7)
with
Λ(λ) =
11−e−τ1λ
τ1λ
1−e−τ1λ
τ1λ−e−τ1λ
11−e−τ2λ
τ2λ
1−e−τ2λ
τ2λ−e−τ2λ
......
...
11−e−τNλ
τNλ
1−e−τNλ
τNλ−e−τNλ
, (2.8)
where yτ1t , . . . , yτN
t are the yields for maturities τ1, . . . ,τN , xt (h) is the forecast ob-
jective, and h the forecast lead, LVt ,SLt ,CVt are latent factors, Λ(λ) is a matrix
of factor loadings, T and c are a matrix and a vector of coefficients, respectively, of
suitable dimensions, εt and ηt are vectors of disturbances with H and Q as respective
variance-covariance matrices. The forecast objective xt (h) is a function of the forecast
horizon, see 2.4 for more details.
Koopman et al. (2010) suggested to make the parameter λ time varying in order
to get a better fit of the yield curve. The parameter λ, or rather its logarithm, is then
included in the state vector and follows joint dynamics with the slope, level, and cur-
vature factors. We also consider a variation of (2.8) in which λ is time-varying, but in
a more parsimonious way by letting λt follow an AR(1). The supervised factor model
is then modified to haveΛ=Λ(λt ). We parameterize the model in the following way
yτ1
t...
yτNt
xt (h)
= Λ
N×3(λt ) 0
N×1
01×3
1
LVt
SLt
CVt
xt (h)
+
ετ1t...
ετNt
0
, (2.9)
50 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
LVt+1
SLt+1
CVt+1
xt+1
log (λt+1)
= c+ T
LVt
SLt
CVt
xt
log (λt )
+ ηt , (2.10)
where T is now the block matrix
T =[
T 00 φ
], (2.11)
and ηt ∼ N (0,Q) with
Q =[
Q 00 σ2
λ
]. (2.12)
The system is now non-linear in the state vector and can be estimated via the extended
Kalman filter1, see for instance Durbin and Koopman (2012). Indicate with αt =[LVt ,SLt ,CVt , xt , l og (λt )]′ the state vector. The measurement equation has then the
form
[yt
xt
]=
[Λ(λt ) 0
0 1
]LVt
SLt
CVt
xt (h)
+
ετ1t...
ετNt
0
= Zt (αt )+[εt
0
], (2.13)
where the state vector αt follows the dynamics specified in equations (2.9)-(2.12).
The extended Kalman filter is based on a local linearization of Zt (αt ) at at |t−1, an
estimate of αt based on the past observations y1, . . . , yt−1 and x1(h), . . . , xt−1(h). The
linearized model is thus[yt
xt
]= Zt (at |t−1)+ Zt (at |t−1)(αt −at |t−1)+
[εt
0
]
= dt + Zt (at |t−1)αt +[εt
0
], (2.14)
where
dt = Zt (at |t−1)− Zt (at |t−1)at |t−1, (2.15)
and
Zt (at |t−1) = ∂Zt (αt )
∂α′t
∣∣∣∣αt=at |t−1
, (2.16)
1Exact estimation procedures for non-linear systems require a major computational effort as opposedto the extended Kalman filter.
2.3. COMPUTATIONAL ASPECTS 51
Zt (αt ) =Λ(λt ) 0 ∂Zt (αt )
∂log (λt )
0 1 0
, (2.17)
with
∂Zt (αt )
∂log (λt )=
e−λt τ1 (λtτ1−eλt τ1+1)
λ2t τ1
λt SLt + e−λt τ1 (τ21λ
2t +λtτ1−eλt τ1+1)
λ2t τ1
λt CVt
...e−λt τN (λtτN−eλt τN +1)
λ2t τN
λt SLt + e−λt τN (τ2Nλ
2t +λtτN−eλt τN +1)
λ2t τN
λt CVt
.
The state-space system is then made up of the measurement equation (2.14) and
state equation (2.10).
2.3 Computational aspects
The objective of the study is to determine the forecasting power of the yield curve
and the supervised factor model (2.1). The forecast performance is based on out-of-
sample forecasts for which a rolling window of fixed size is used for the estimation of
the parameters. The log-likelihood is maximized for each estimation window.
Estimation method
The parameters of the state-space model are estimated by maximum likelihood. The
likelihood is delivered by the Kalman filter. We employ the univariate Kalman filter as
derived in Koopman and Durbin (2000) as we assume a diagonal covariance matrix
for the innovations in the measurement equation. The maximum of the likelihood
function has no explicit form solution and numerical methods have to be employed.
We make use of the following two algorithms.
• CMA-ES. Covariance Matrix Adaptation Evolution Strategy, see Hansen and
Ostermeier (1996)2. This is a genetic algorithm that samples the parameter
space according to a Gaussian search distribution which changes according to
where the best solutions are found in the parameter space;
• BFGS. Broyden-Fletcher-Goldfarb-Shanno, see for instance Press et al. (2002).
This algorithm belongs to the class of quasi-Newton methods. The algorithm
needs the computation of the gradient of the function to be minimized.
The CMA-ES algorithm is used for the first estimation window for each forecast target.
This algorithm is particularly useful, in this context, if no good guess of initial values
is available. We then use the BFGS algorithm for the rest of the estimation windows
2See https://www.lri.fr/~hansen/cmaesintro.html for references and source codes. The au-thors provide C source code for the algorithm which can be easily converted into C++ code.
52 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
as this method is substantially faster than the CMA-ES but more dependent on initial
values. We use algorithmic (or automatic) differentiation3 to compute gradients. We
make use of the ADEPT library C++ library, see Hogan (2013)4. The advantage of using
algorithmic differentiation over finite differences is twofold: increased speed and
elimination of approximation errors.
Speed improvements
To gain speed we chose C++ as the programming language, using routines from
the Numerical Recipes, Press et al. (2002) 5. We compile and run the executables
on a Linux 64-bit operating system using GCC 6. We use Open MPI 1.6.4 (Message
Passing Interface) with the Open MPI C++ wrapper compiler mpic++ to parallelise the
maximum likelihood estimations 7. We compute gradients using the ADEPT library
for algorithmic differentiation, see Hogan (2013).
2.4 Empirical application
Data
The macroeconomic variables selected as forecast objectives are: consumer price
index (CPI), personal consumer expenditures (PCE), producer price index (PPI), real
disposable income (RDI), unemployment rate (UR), and industrial production (IP).
The macroeconomic data have been taken from FRED (Federal Reserve Economic
Data)8.
Yield curve
We use two datasets for the yield-curve data. The first one is from Gürkaynak et al.
(2007)9. We skip-sample the data in order to avoid inducing artificial persistence in
the time series. We take the first yield registered in the month as the yield for that
month. This is consistent with the macroeconomic variables whose values refer to
3See for instance Verma (2000) for an introduction to algorithmic differentiation.4For a user guide see http://www.cloud-net.org/~clouds/adept/adept_documentation.
pdf.5See Aruoba and Fernández-Villaverde (2014) for a comparison of different programming languages
in economics and Fog (2006) for many suggestions on how to optimize software in C++.6See http://gcc.gnu.org/onlinedocs/ for more information on the Gnu Compiler Collection,
GCC.7See http://www.open-mpi.org/ for more details on Open MPI and Karniadakis (2003) for a review
of parallel scientific computing in C++ and MPI.8The data can be downloaded from the website of the Federal Reserve Bank of St. Louis: http://
research.stlouisfed.org/fred2, Help: http://research.stlouisfed.org/fred2/help-faq.9The Gürkaynak, Sack and Wright dataset can be downloaded from http://www.federalreserve.
gov/pubs/feds/2006.
2.4. EMPIRICAL APPLICATION 53
the first days of the month. The second dataset is the yield-curve data from Diebold
and Li (2006). All data refer to the US economy.
Macroeconomic variables
• CPI. Series ID: CPIAUCSL, Title: Consumer Price Index for All Urban Con-
sumers: All Items, Source: U.S. Department of Labor: Bureau of Labor Statis-
tics, Release: Consumer Price Index, Units: Index 1982-84=100, Frequency:
Monthly, Seasonal Adjustment: Seasonally Adjusted, Notes: Handbook of Meth-
ods (http://www.bls.gov/opub/hom/pdf/homch17.pdf).
• PPI. Series ID: PPIFGS, Title: Producer Price Index: Finished Goods, Source: U.S.
Department of Labor: Bureau of Labor Statistics, Release: Producer Price Index,
Units: Index 1982=100, Frequency: Monthly, Seasonal Adjustment: Seasonally
Adjusted, Series ID: UNRATE, Title: Civilian Unemployment Rate, Source: U.S.
Department of Labor: Bureau of Labor Statistics, Release: Employment Situ-
ation, Units: Percent, Frequency: Monthly, Seasonal Adjustment: Seasonally
Adjusted.
• RDI. Series ID: DSPIC96, Title: Real Disposable Personal Income, Source: U.S.
Department of Commerce: Bureau of Economic Analysis, Release: Personal In-
come and Outlays, Units: Billions of Chained 2009 Dollars, Frequency: Monthly,
Seasonal Adjustment: Seasonally Adjusted Annual Rate, Notes: BEA Account,
Code: A067RX1, A Guide to the National Income and Product Accounts of
the United States (NIPA) - (http://www.bea.gov/national/pdf/nipaguid.
pdf).
• UR. Series ID: UNRATE, Title: Civilian Unemployment Rate, Source: U.S. De-
partment of Labor: Bureau of Labor Statistics, Release: Employment Situation,
Units: Percent, Frequency: Monthly, Seasonal Adjustment: Seasonally Adjusted.
• IP. Series ID: INDPRO, Title: Industrial Production Index, Source: Federal Re-
serve Economic Data, Units: Levels, Frequency: Monthly, Index: 2007=100, Sea-
sonal Adjustment: Seasonally Adjusted, Link: http://research.stlouisfed.
org/fred2.
Competing models
We choose different competing models diffusely used in the forecasting literature, in
order to assess the relative forecasting performance of the supervised factor model.
We divide these models into direct multi-step and indirect (recursive) forecasting
models.
54 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
Direct forecasting models
The first model is the following restricted AR(p) process
xt+h(h) = c +φ1xt (h)+ . . .φp xt−p (h)+εt+h . (2.18)
The second model is a restricted M A(q) process
xt+h(h) = c +θ1εt + . . .θqεt−q +εt+h . (2.19)
Both models are estimated by maximum likelihood. The lags p and q are selected
for each estimation sample as the values that minimize the Bayesian information
criterion. In particular, we consider p, q ∈ 1,2,3.
The third model is principal component regression (PCR). In the first step, princi-
pal components are extracted from the regressors Yt = [yτ1t , . . . , yτN
t , xt (h)]; xt+h(h) is
then regressed on them to obtain βPC R for time indexes 1 ≤ t ≤ Ti −h. In the second
step, the principal components are projected at time Ti and then multiplied by βPC R
to obtain the h-period ahead forecast. We estimate three factors.
The fourth model considered is partial least squares regression (PLSR). In the first
step, the partial least squares components xmt are computed using the forecast target
xt (h) : h ≤ t ≤ Ti and the predictors Yt = [yτ1t , . . . , yτN
t , xt (h)] with 1 ≤ t ≤ Ti −h
where M ≤ (N +1) is the number of partial least squares components and N +1 is
the number of predictors including the lagged value of the forecast objective. In the
second step, the partial least squares components xmt are regressed on the predic-
tors Yt to recover the coefficient vector βPLS . Note that as the partial least squares
components are a linear combination of the regressors, the relation is exact, i.e. the
residuals from this regression are (algebraically) null. In the third step, the partial
least squares components are projected at time Ti by multiplying YTi by βPLS . The
projected PLS components at time Ti are then summed to obtain the h-period ahead
forecast xTi+h(h) =∑Mm=1 xTi (h)m . We estimate three PLS directions.
The fifth direct forecasting method considered is a two-step procedure as de-
scribed in equations (2.5) in which the factors are extracted using the Kalman filter
and the factor loadings have the NS structure. We refer to these factor models as
unsupervised.
The sixth model is a two-step procedure where the factors are first extracted using
the semi-parametric factor model implemented in Härdle, Majer, and Schienle (2012)
and then used to forecast with an auxiliary equation as in eqn. (2.3).
The seventh competing method is the forecast combination (CF-NS) model de-
rived in Hillebrand et al. (2012).
Finally, we compare the forecast performance of the supervised models with fixed
λ (2.7), (2.8), and time varying λ (2.14), (2.10) to their unsupervised counterparts. In
these specifications the factors are first extracted using the Kalman filter, the forecasts
are then obtained using the forecast equation
xt+h(h) = c + ftβ+γxt (h)+ut , (2.20)
2.4. EMPIRICAL APPLICATION 55
where c, β, and γ are parameters to be estimated, ut is the error term, and ft is the
vector of filtered factors.
Indirect forecasting models
The first model is the following AR(p) process
xt+h(h) = c +φ1xt+h−1(h)+ . . .φp xt+h−p (h)+εt+h . (2.21)
The second model is an M A(q) process
xt+h(h) = c +θ1εt+h−1 + . . .θqεt+h−q +εt+h . (2.22)
Both models are estimated using maximum likelihood. The lags p and q are selected
for each estimation sample as the values that minimize the Bayesian information
criterion. In particular, we consider p, q ∈ 1,2,3.
Forecasting
Forecast objective
We stationarize the forecast objectives by treating them as either I (2) or I (1) variables.
In particular we follow Stock and Watson (2002b) and treat price indexes as I (2)
processes and real variables as I (1) and take the following transformations
xt (h) =k(log (X t )− log (X t−h))/h if X t is I(1),
k(log (X t )− log (X t−h))/h −k(log (X t−h − log (X t−h−1)) if X t is I(2),(2.23)
where k is a scaling factor. In particular, we have k = 1200, h ∈ 1,3,6,9,12, X t ∈C PIt ,PC Et ,PPIt ,RD It ,U Rt .
Forecasting scheme
The aim is to compute the forecast of the macro variable xt (h) at time t +h, i.e.
xt+h(h) where h is the forecast lead. We consider a rolling windows scheme. The
reason is that one of the requirements for the application of the Giacomini and White
(2006) test, in case of nested models, is to use rolling windows. We build series of
forecast errors of length S for all forecast objectives/leads. The complete time series
is indexed Yt : t ∈N>0, t ≤ T where T is the sample length of the complete dataset
and Yt = y1t , ..., y N
t , xt (h), where y it , i = 1,2, ..., N . The estimation sample takes into
account observations indexed Yt : t ∈N>0,Ti −R +1 ≤ t ≤ Ti for i ∈N>0, i ≤ S with
T1 = R = T−S−hmax+1 the index of the last observation of the first estimation sample,
which coincides with the size of the rolling window, and Ti = T1 + i for i ∈N>0, i ≤ S
56 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
and hmax is the maximum forecast lead. In particular, we chose different values of S,
and R depending on the complete sample length of the yield curve data, T . We chose
hmax = 12 for all applications. The forecasting strategy for the indirect h-step ahead
forecasts for the supervised factor model, and indirect forecasts (equations (2.7) and
(2.8)) is the following:
(i) estimate the system parameters of the dynamic factor model using information
from time Ti −R +1 up to time Ti by maximizing the log-likelihood function
delivered by the Kalman filter;
(ii) indicating withαt the state vector containing the latent factors and the forecast
target, compute the smoothed estimate at time Ti of the state vector, i.e. αTi ;
(iii) iterate h times on the filtered state at time Ti , αTi , using the estimated parame-
ters to obtain the forecast:
xT+h|T = [01×k : 1]
ThαTi +
h−1∑j=0
Tjc
; (2.24)
for i = 1, . . . ,S.
Test of forecast performance
We use the conditional predictive ability (CPA) test proposed in Giacomini and White
(2006)10 using a quadratic loss function. This test also allows to compare nested
models, provided a rolling windows scheme for parameter estimation is used. The
autocorrelations of the loss differentials are taken into account by computing Newey
and West (1987) standard errors. We follow the “rule of thumb” in Clark and Mc-
Cracken (2011) and take a sample split ratio π= SR approximately equal to one.
Results
To assess the relative forecasting performance of the supervised factor models with
respect to the competing methods, we present mean squared prediction errors ratios
between forecasts from model (2.7) and the competing models. The results are sum-
marised in tables 2.1-2.9 in Appendix 2.8. In bold are the ratios lower than 1 which
indicate a better forecasting performance of the supervised factor model, eqn. (2.7),
compared to the competing method. We consider three applications:
(i) Gürkaynak et al. (2007) yield curve data, 7 maturities corresponding to 1, 2, 3,
4, 5, 6, and 7 years, sample size T = 617, rolling window of size R = T1 = 306,
10At http://www.runshare.org/CompanionSite/site.do?siteId=116 the authors provideMATLAB codes for the test.
2.4. EMPIRICAL APPLICATION 57
number of forecasts S = 300. Observations range from August 1961 to Decem-
ber 2012. The 1-step ahead forecasts range from February 1987 to January 2012.
The 12-step ahead forecasts range from January 1988 to December 2012.
(ii) Gürkaynak et al. (2007) yield curve data, 30 maturities corresponding to 1,
2, ..., 29, and 30 years, sample size T = 325, rolling window of size R = T1 =189, number of forecasts S = 125. Observations range from December 1985
to December 2012. The 1-step ahead forecasts range from September 2001 to
January 2012. The 12-step ahead forecasts range from August 2002 to December
2012.
(iii) Diebold and Li (2006) yield curve data, 17 maturities corresponding to 3, 6,
9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120 months, sample
size T = 346, rolling window of size R = T1 = 185, number of forecasts S = 150.
Observations range from February 1972 to November 2000. The 1-step ahead
forecasts range from July 1987 to December 1999. The 12-step ahead forecasts
range from June 1988 to November 2000.
In the tables we label the different forecasting models according to the following
convention.
• model 1. Principal component regression (PCR);
• model 2. Partial least squares regression (PLSR);
• model 3. AR(p) direct, equation (2.18);
• model 4. MA(q) direct, equation (2.19);
• model 5. AR(p) indirect, equation (2.21);
• model 6. MA(q) indirect, equation (2.22);
• model 7. Supervised dynamic factor model with Nelson-Siegel factor loadings
(2.8), and direct forecasts (2.2);
• model 8. Supervised dynamic factor model with Nelson-Siegel loadings (2.7),
and indirect forecasts (2.8);
• model 9. Supervised dynamic factor model with Nelson-Siegel loadings with
time varying λ=λ(t ), and indirect forecasts (2.10), (2.9);
• model 10. Unsupervised dynamic factor model with Nelson-Siegel loadings,
(2.7), (2.8), and forecast equation 2.20;
• model 11. Unsupervised dynamic factor model with Nelson-Siegel loadings
and time varying λ=λ(t ), (2.10), (2.9), and forecast equation 2.20;
58 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
• model 12. Factors extracted using the semiparametric factor model, Härdle
et al. (2012), and forecast equation 2.20;
• model 13. Forecast combination-Nelson Siegel (CF-NS), see Hillebrand et al.
(2012).
Looking at tables 2.1-2.9 in the Appendix, we can make the following remarks (divided
with respect to the three applications):
(i) Concerning the first application we note that the supervised factor model (2.7)
delivers better forecasts for output variables, namely, real disposable income
(RDI), industrial production (IP), and unemployment rate (UR), relative to
the different inflation variables, that is, consumer price index (CPI), personal
consumption expenditures (PCE), and producer price index (PPI). With regard
to the performance relative to other forecasting schemes the supervised factor
model generally performs similarly to or better than principal components
regression (model 1), partial least squares regression (model 2), a two-step fore-
casting procedure (model 12), the unsupervised scheme (model 10), and the
CF-NS procedure (model 13), as can be seen from tables 2.1-2.3. For inflation
related variables the direct and indirect MA(q) and AR(p) forecasts are hard to
beat, in particular for forecast lead h = 1. The supervised factor model performs
generally better than the unsupervised counterpart. Allowing for dynamics in
the λ improves the forecasts only for few forecast targets.
For real disposable income (RDI), the performance of model 8, relative to uni-
variate models, in the first half of the forecasting sample (corresponding to the
first 13 years of the Greenspan monetary regime, 1987-2000), is considerably
worse than that in the second half of the forecasting sample (corresponding to
the last 6 years of the Greenspan monetary regime and the first 6 years of the
Bernanke monetary regime, 2001-2012) as can be seen from tables 2.2 and 2.3.
(ii) The results for the second application are similar to the ones for the first appli-
cation, as can be seen from tables 2.4-2.6. For real disposable income (RDI), the
performance of model 8, relative to univariate models, in the first half of the
forecasting sample (corresponding to the last 6 years of the Greenspan mon-
etary regime, 2001-2006), is considerably worse than that in the second half
of the forecasting sample (corresponding to the first 6 years of the Bernanke
monetary regime, 2006-2012) as can be seen from tables 2.5 and 2.6.
(iii) Regarding the third application the differences between inflation and output
variables is less marked, as can be seen from tables 2.7-2.9. The supervised
framework delivers forecasts similar to or better than the unsupervised frame-
works. Unemployment rate (UR) and industrial production (IP) are the two
variables for which supervision is more beneficial. Allowing for dynamics in the
λ parameter does not improve the forecasts. For real disposable income (RDI),
2.5. CONCLUSIONS 59
the performance of model 8, relative to univariate models, in the first half of
the forecasting sample (corresponding to the first 6 years of the Greenspan
monetary regime, 1987-1993), is better than that in the second half of the fore-
casting sample (corresponding to the Greenspan monetary regime going from
1994-1999) as can be seen from tables 2.8 and 2.9.
Overall, from tables 2.1-2.9 we can see that the yield curve has more predictive power
for the periods 1987-1994 (the early Greenspan monetary regime) and 2006-2012
(the early Bernanke monetary regime) as compared to the period 1994-2006 (the late
Greenspan monetary regime). The good forecasting power of the yield curve in the
early Greenspan period is consistent with the findings in Giacomini and Rossi (2006).
Note that in this study we cannot make use of the supervision measure proposed in
Boldrini and Hillebrand (2015) as the observable variables are not guaranteed to be
stationary.
2.5 Conclusions
In this paper we study the forecasting power of the yield curve for some macroecomic
variables, for the US economy, in the framework of a supervised factor model. In this
model the factors are extracted conditionally on the forecast target. The model has a
linear state-space representation and standard Kalman filtering techniques apply.
We forecast macroeconomic variables using factors extracted from the yield curve.
We use the yield curve data from Gürkaynak et al. (2007) and Diebold and Li (2006)
and macroeconomic data from FRED. We use the Nelson and Siegel factor load-
ings and allow for dynamics in the factors. We forecast consumer price index (CPI),
personal consumer expenditures (PCE), producer price index (PPI), real disposable
income (RDI), unemployment rate (UR), and industrial production (IP).
We find that supervising the factor extraction can improve the forecasting perfor-
mance of the factor model. For this dataset and specification the supervised factor
model outperforms principal components regression, and partial least squares regres-
sion on most targets, in particular for UR, RDI, and IP. In forecasting inflation, both
measured by consumer price index, and producer price index, M A(q) and AR(p)
processes are difficult to beat, especially for one-step ahead forecasts. Allowing for
dynamics in the Nelson and Siegel factor loadings generally does not improve the
forecasts. The supervised factor model performs particularly well in forecasting un-
employment rate and real disposable income. Furthermore, supervising the factor
extraction leads in most cases to improved forecasts compared to unsupervised two-
step forecasting schemes.
We find that the yield curve has forecast power for unemployment rate, real
disposable income, and industrial production but less so for inflation measures. Sim-
ilarly to Giacomini and Rossi (2006), Rudebusch and Williams (2009), and Stock and
Watson (1999a) we find that the predictive ability of the yield curve is somewhat
60 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE
unstable and has changed through the years. In particular, we find that the yield
curve has more predictive power for the periods 1987-1994 (the early Greenspan
monetary regime) and 2006-2012 (the early Bernanke monetary regime) as compared
to the period 1994-2006 (the late Greenspan monetary regime). The good forecasting
power of the yield curve in the early Greenspan period is consistent with the findings
in Giacomini and Rossi (2006).
2.6 Acknowledgements
This research was supported by the European Research Executive Agency in the
Marie-Sklodowska-Curie program under grant number 333701-SFM.
2.7. REFERENCES 61
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2.8. APPENDIX 65
2.8 Appendix
Tables
In this section we report mean square forecast errors (MSFE) ratios corresponding to
the empirical application (see Section 2.4). The results correspond to MSFE ratios
between model 8 and the competing models (see Section 2.4 for the description of
the different models involved). We consider different subsamples of the dataset. In
the tables below, three, two, and one stars refer to significance levels 0.01, 0.05, and
0.10 for the null hypothesis of equal conditional predictive ability for the Giacomini
and White (2006) test.
66 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVETab
le2.1.M
SFE
ratios
for
wh
ole
forecastsam
ple
(Gü
rkaynak
etal.(2007)yield
data,7
matu
rities).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11,02
1,021,14
1,37*1,14
1,37*1
1,011
1,011,02
0,99*3
1,161,16
1,121,23
1,121,31**
1,141,04
1,151,16
1,161,04
CP
I6
1,031,03
1,051,07
1,021,03
11,01
1,031,03
1,030,99
90,98
0,981,01
1,021
10,98**
0,960,97**
0,98*0,98
0,96**12
10,99
11
1,01*1,01*
11,01
1,010,99
10,96**
11,01
11,33***
1,69***1,33***
1,69***1*
0,81*1*
0,991,01
0,6**3
11
10,99
0,951,01
10,88***
10,99
11
PC
E6
0,990,99
0,990,99
11
10,95*
0,990,98
0,991
90,98
0,980,98
0,981
10,99
10,97**
0,98*0,98
0,9912
1,011,01
1,021,03
11
11
11
1,010,99
11,02
1,011,14
1,29*1,14
1,29*1
0,131
1,011,02
0,82***3
1,061,06
1,031,1
1,041,18
1,041,03
1,041,05
1,061,02
PP
I6
1,071,07
1,071,13*
1,021,02
1,031,01
1,051,07
1,070,99
91,01
11,02
1,011*
1*1
10,98
1,011,01
0,9612
1,051,05
1,071,08*
11
1,010,99*
1,021,04
1,050,96
11
11,01
1,021,01
1,021
0,931
11
0,973
1,081,08
1,14*1,18*
1,071,16*
1,11,01
1,111,11
1,081,06**
RD
I6
0,890,89
1,021,02
1,011,03
0,910,79**
0,950,92
0,891
90,83*
0,84*0,98
0,971
1,030,87**
0,76**0,94
0,890,84*
0,9912
0,73**0,75**
0,950,96
1,061,06
0,81*0,79*
0,860,82*
0,73**1
10,97
0,971,12**
1,07**1,12**
1,07**1
0,79***1
0,990,98
1,07***3
0,950,95
1,1*0,96
10,76
0,94**0,85
0,95**0,96
0,950,82**
UR
60,9
0,90,98
0,960,99**
0,65**0,8**
0,920,84*
0,910,9
0,65***9
0,810,81
0,83**0,89
0,81**0,68***
0,75**1,08
0,75**0,86
0,820,65***
120,81**
0,81**0,82***
0,91**0,74**
0,77***0,73**
0,850,72**
0,87*0,81**
0,72**
10,95
0,941,07
1,041,07
1,040,83**
0,991
0,980,95
13
0,84**0,85**
0,990,96
0,910,68
0,6**0,51***
0,9**0,88*
0,85**0,66*
IP6
0,83**0,84**
0,991,11
0,950,86
0,7**0,99
0,86***0,88
0,84**0,8*
90,96**
0,97**1,13
1,31,06
1,210,88**
0,880,95***
1,030,97**
1,0812
1,091,1
1,331,28
1,081,52
1,02**1,22
1,061,18
1,11,36
Gü
rkaynak
etal.(2007)
yieldd
ata.7m
aturities,fro
m1
to7
years.MSF
Eratio
sb
etween
mo
del8
and
mo
dels
1−13
for
CP
I,PC
E,P
PI,R
DI,U
R,an
dIP
for
allforecastin
glead
sh
.Avalu
elow
erth
anon
ein
dicates
alow
erM
SFEofm
odel8
w.r.t.m
odels
1−13.O
ne,tw
o,and
three
starsm
ean.10,.05,an
d.01
statisticalsignifi
cance,resp
ectively,for
the
Giacom
inian
dW
hite
(2006)testw
ithq
uad
raticloss
fun
ction.T
he
nu
mb
erofforecasts
isS=
300.Th
e1-step
ahead
forecastsran
gefrom
Febru
ary1987
toJan
uary
2012.Th
e12-step
ahead
forecasts
range
from
Janu
ary1988
toD
ecemb
er2012.T
he
forecast
perio
dco
rrespo
nd
sto
part
ofth
eG
reensp
an(1987-2006)
and
Bern
anke
(2006-2012)m
on
etaryregim
es.
2.8. APPENDIX 67Ta
ble
2.2.
MSF
Era
tio
sfo
rfi
rsth
alfo
ffo
reca
stsa
mp
le(G
ürk
ayn
aket
al.(
2007
)yi
eld
dat
a,7
mat
uri
ties
).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11
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1,12
*1,
25**
*1,
12*
1,25
***
11
10,
991
0,84
**3
0,99
0,99
11,
041,
011,
031,
010,
991
0,99
0,99
0,98
CP
I6
0,99
0,98
1,06
1,07
1,01
1,01
0,94
1**
0,97
0,97
0,99
0,91
90,
90,
91,
021,
011,
011,
010,
93**
10,
90,
890,
90,
86*
120,
85*
0,85
*1,
021,
031,
011,
010,
92**
*1,
020,
83*
0,85
*0,
85*
0,81
**
11,
011,
011,
21**
1,83
***
1,21
**1,
83**
*1
0,88
*1
0,98
1,01
0,59
*3
0,99
0,99
0,99
0,98
0,91
1,02
0,99
10,
990,
970,
991*
PC
E6
0,99
0,99
11
11
10,
9**
10,
980,
990,
999
0,98
0,98
0,99
0,98
11
0,99
10,
980,
970,
980,
9912
1,02
1,02
1,04
**1,
05**
1,01
1,01
1,01
1,01
1,01
0,99
1,02
0,99
*
11
11,
081,
231,
081,
231
0,94
11*
10,
863
1,03
1,03
1,03
1,04
11,
11*
1,01
0,98
**1,
011,
031,
030,
97P
PI
61,
021,
021,
021,
031
10,
990,
99*
0,99
1,02
1,02
0,96
90,
980,
981
11
10,
990,
990,
920,
970,
980,
9412
0,97
0,97
11
11
0,99
**0,
950,
890,
970,
970,
93
10,
990,
991,
021,
021,
021,
021
0,96
10,
990,
990,
953
0,95
0,95
1,15
1,17
1,04
1,14
1,01
1,02
1,01
1,01
0,95
1,07
RD
I6
0,79
**0,
8*1,
141,
151,
091,
130,
841,
020,
880,
87*
0,8*
19
0,75
**0,
75**
1,17
1,15
1,12
1,2
0,8*
*0,
990,
87*
0,87
*0,
75**
1,01
120,
63**
0,63
**1,
16*
1,11
**1,
161,
24**
0,72
*1,
010,
75*
0,79
*0,
63**
1**
11,
02*
1,02
*1,
051,
051,
051,
051
0,9*
*1
1,04
**1,
02**
1,05
**3
11
1,17
**1,
051
0,96
0,92
1,06
***
0,92
1,03
1,01
0,99
*U
R6
0,82
0,82
10,
880,
98*
0,73
0,65
10,
650,
840,
830,
799
0,61
0,61
0,72
*0,
770,
820,
59**
0,54
1,07
**0,
470,
670,
620,
6812
0,52
*0,
52*
0,64
*0,
720,
830,
56**
0,51
*1,
28*
0,39
0,62
0,53
*0,
66**
10,
95**
*0,
96**
*1
1,05
11,
050,
971,
01*
11*
**0,
96**
*1,
22**
30,
850,
851,
071,
091
0,93
0,97
0,94
0,91
0,94
0,85
1,17
IP6
0,74
0,74
1,05
1,05
1,01
0,92
0,95
1,04
0,77
0,86
0,74
1,31
**9
0,62
0,62
0,94
**0,
99*
0,99
**0,
840,
911,
070,
610,
740,
621,
34**
*12
0,61
*0,
61*
0,91
0,62
1,14
0,89
1,02
1,52
**0,
59*
0,77
*0,
61*
1,62
*
Gü
rkay
nak
etal
.(20
07)
yiel
dd
ata.
7m
atu
riti
es,f
rom
1to
7ye
ars.
MSF
Era
tio
sb
etw
een
mo
del
8an
dm
od
els
1−1
3fo
rC
PI,
PC
E,P
PI,
RD
I,U
R,a
nd
IPfo
ral
lfo
reca
stin
gle
ads
h.A
valu
elo
wer
than
one
ind
icat
esa
low
erM
SFE
ofm
odel
8w
.r.t
.mod
els
1−1
3.O
ne,
two,
and
thre
est
ars
mea
n.1
0,.0
5,an
d.0
1st
atis
tica
lsig
nifi
can
ce,r
esp
ecti
vely
,fo
rth
eG
iaco
min
ian
dW
hit
e(2
006)
test
wit
hq
uad
rati
clo
ssfu
nct
ion
.Th
en
um
ber
off
ore
cast
sis
S′ =
150
(th
efi
rst
hal
foft
he
S=
300
com
ple
tefo
reca
stsa
mp
le).
Th
e1-
step
ahea
dfo
reca
sts
ran
gefr
omFe
bru
ary
1987
toJu
ly19
99.T
he
12-s
tep
ahea
dfo
reca
sts
ran
gefr
omJa
nu
ary
1988
toJu
ne
2000
.Th
efo
reca
stp
erio
dco
rres
pon
ds
top
art
oft
he
Gre
ensp
an(1
987-
2006
)m
on
etar
yre
gim
e.
68 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVETab
le2.3.M
SFE
ratios
for
secon
dh
alfoffo
recastsamp
le(G
ürkayn
aketal.(2007)
yieldd
ata,7m
aturities).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11,02
1,021,14
1,391,14
1,391
1,021
1,021,02
1,033
1,191,19
1,151,27
1,141,37**
1,171,05
1,181,19
1,191,06
CP
I6
1,041,04
1,051,07
1,021,03
1,021,01
1,041,04
1,041,01
91
0,991,01
1,021*
10,98**
0,95*0,98*
0,99*1
0,9812
1,031,02
0,990,99
1,011,01*
1,011,01
1,051,02*
1,030,99
11
11,46**
1,571,46**
1,571
0,761
11
0,61*3
1,021,01
1,011,01
0,991,01
1,010,77***
1,011,01
1,020,99
PC
E6
0,990,99
0,990,99
1,011,01
11,01
0,980,98
0,991
90,98
0,980,98*
0,981
10,99
0,990,97**
0,980,98
0,9912
0,990,99
11
0,990,99
0,991
0,981
0,990,98
11,02
1,021,16
1,311,16
1,311
0,111
1,021,02
0,82**3
1,071,06
1,031,11
1,051,2
1,041,03
1,051,06
1,071,03
PP
I6
1,091,08
1,081,15*
1,021,02
1,041,02
1,071,08
1,091
91,01
1,011,02
1,011**
1**1
10,99
1,011,01
0,9712
1,061,06
1,091,1**
11
1,011
1,041,06
1,060,97
11,01
1,011
1,031
1,031
0,911
1,011,01
0,993
1,191,19
1,141,18
1,081,17
1,161,01
1,191,19
1,191,05*
RD
I6
11
0,930,92
0,950,95
0,990,65***
1,020,98
11
90,94
0,950,83
0,830,89*
0,90,95
0,61***1,01
0,910,94
0,96*12
0,890,93
0,790,84
0,970,91
0,920,63***
1,030,86
0,891,01
10,94
0,931,2**
1,08**1,2**
1,08**1
0,72***1
0,950,94
1,09***3
0,930,93
1,070,91
10,68
0,95**0,77*
0,96**0,93
0,930,74**
UR
60,93
0,940,97
11
0,620,89
0,880,95**
0,950,94
0,6***9
0,890,89
0,870,93
0,81**0,71**
0,84*1,08
0,89*0,93
0,890,65**
120,91
0,920,87**
0,97**0,73*
0,85**0,81*
0,79**0,88
0,950,92
0,74**
10,94
0,941,12
1,031,12
1,030,77**
0,981
0,960,94
0,913
0,84*0,85*
0,95*0,89
0,870,6
0,5***0,41***
0,89**0,86
0,84*0,54**
IP6
0,870,87
0,971,14
0,930,84
0,65**0,98
0,9**0,89
0,870,72***
91,07*
1,071,17
1,381,07
1,310,88**
0,861,06**
1,11,07
1,04**12
1,251,25
1,441,52
1,071,7
1,02***1,19**
1,21,29
1,251,32
Gü
rkaynak
etal.(2007)
yieldd
ata.7m
aturities,fro
m1
to7
years.MSF
Eratio
sb
etween
mo
del8
and
mo
dels
1−13
for
CP
I,PC
E,P
PI,R
DI,U
R,an
dIP
for
allforecastin
glead
sh
.Avalu
elow
erth
anon
ein
dicates
alow
erM
SFEofm
odel8
w.r.t.m
odels
1−13.O
ne,tw
o,and
three
starsm
ean.10,.05,an
d.01
statisticalsignifi
cance,resp
ectively,fo
rth
eG
iacom
inian
dW
hite
(2006)testw
ithq
uad
raticlo
ssfu
nctio
n.T
he
nu
mb
ero
fforecasts
isS ′=
150(th
eseco
nd
halfo
fthe
S=300
com
plete
forecastsam
ple).T
he
1-stepah
eadfo
recastsran
gefro
mA
ugu
st1999to
Janu
ary2012.T
he
12-stepah
eadfo
recastsran
gefro
mJu
ly2000
toD
ecemb
er2012.T
he
forecastp
eriod
corresp
on
ds
top
artofth
eG
reensp
an(1987-2006)
and
Bern
anke
(2006-2012)m
on
etaryregim
es.
2.8. APPENDIX 69Ta
ble
2.4.
MSF
Era
tio
sfo
rw
ho
lefo
reca
stsa
mp
le(G
ürk
ayn
aket
al.(
2007
)yi
eld
dat
a,30
mat
uri
ties
).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11,
021
1,11
1,39
1,11
1,39
11
10,
991,
021,
093
1,2
1,18
1,17
1,28
1,15
1,44
**0,
711
1,21
11,
21,
1C
PI
61,
081,
061,
081,
091,
061,
060,
810,
991,
080,
991,
081,
079
1,03
1,02
0,95
0,92
1,01
*1,
010,
691,
011*
11,
031,
0212
1,04
**1,
03**
0,95
0,89
1,02
1,02
0,81
1,01
1,08
*0,
99*
1,05
**1,
02
11,
011,
011,
57*
1,86
1,57
*1,
861
0,98
1,01
0,98
1,01
0,61
31,
011,
011,
011,
010,
991
0,96
*0,
871,
010,
911,
011
PC
E6
0,97
0,97
0,97
0,96
1,01
1,01
0,98
0,98
0,95
10,
971,
019
0,99
11
11
10,
95*
0,95
0,96
0,97
0,99
112
0,97
0,97
0,88
0,96
11
0,92
1,04
0,95
0,99
0,97
1
11,
031,
021,
091,
251,
091,
251
11
0,99
1,02
0,83
*3
1,07
1,06
1,03
1,13
1,02
1,22
0,68
**0,
991,
060,
981,
071,
06P
PI
61,
091,
091,
071,
171,
051,
040,
870,
981,
050,
971,
091,
059
1,02
1,03
0,98
1,01
1,01
1,01
0,76
10,
961
1,02
1,01
121,
071,
081,
06*
1,14
**1,
021,
020,
840,
991,
031,
021,
071,
02
11,
021,
020,
991,
080,
991,
081
0,99
11,
061,
021,
013
1,22
*1,
24*
1,18
1,21
1,13
1,3
1,04
0,98
1,21
11,
22*
0,97
***
RD
I6
0,96
0,98
0,94
0,95
0,96
0,99
0,75
**1,
021,
011,
010,
970,
67**
*9
0,79
*0,
830,
750,
790,
860,
870,
29*
1,01
0,8*
*1
0,8*
0,54
**12
0,66
**0,
69**
0,78
0,83
0,88
0,89
0,99
***
0,99
0,68
***
0,99
0,66
**0,
48
10,
951,
061,
14*
0,98
1,14
*0,
981
10,
99**
0,93
0,95
0,99
*3
0,93
0,98
0,95
0,78
0,88
0,62
0,37
***
0,96
0,94
1,04
0,9
0,67
UR
60,
970,
970,
931,
220,
90,
640,
34**
0,92
0,94
0,94
**0,
960,
79
0,94
*0,
930,
890,
990,
810,
780,
24*
1,01
0,88
0,96
0,95
*0,
8712
1,06
*1,
05*
0,96
1,18
0,77
1,01
*0,
440,
930,
910,
931,
071,
1
10,
991,
061,
110,
961,
110,
961
1,04
11
0,98
0,98
30,
960,
950,
990,
920,
960,
650,
8*0,
980,
960,
980,
970,
66IP
60,
890,
870,
931,
310,
930,
930,
781,
020,
880,
970,
91,
029
1,14
1,1
1,15
1,56
1,06
1,46
0,79
1,02
0,98
0,97
1,15
*1,
6512
1,37
1,27
1,44
1,66
1,04
1,84
0,9
1,03
1,04
0,96
1,36
2,13
Gü
rkay
nak
etal
.(20
07)
yiel
dd
ata.
30m
atu
riti
es,f
rom
1to
30ye
ars.
MSF
Era
tio
sb
etw
een
mo
del
8an
dco
mp
etin
gm
od
els
for
CP
I,P
CE
,PP
I,R
DI,
UR
,an
dIP
for
all
fore
cast
ing
lead
sh
.Ava
lue
low
erth
ano
ne
ind
icat
esa
low
erM
SFE
ofm
od
el8
w.r
.t.m
od
els
1−1
3.O
ne,
two,
and
thre
est
ars
mea
n.1
0,.0
5,an
d.0
1st
atis
tica
lsig
nifi
can
ce,
resp
ecti
vely
,fo
rth
eG
iaco
min
ian
dW
hit
e(2
006)
test
wit
hq
uad
rati
clo
ssfu
nct
ion
.Th
en
um
ber
off
ore
cast
sis
S=
125.
Th
e1-
step
ahea
dfo
reca
sts
ran
gefr
om
Sep
tem
ber
2001
toJa
nu
ary
2012
.Th
e12
-ste
pah
ead
fore
cast
sra
nge
fro
mA
ugu
st20
02to
Dec
emb
er20
12.T
he
fore
cast
per
iod
corr
esp
on
ds
top
arto
fth
eG
reen
span
(198
7-20
06)a
nd
Ber
nan
ke(2
006-
2012
)m
on
etar
yre
gim
es.
70 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVETab
le2.5.M
SFE
ratios
for
firsth
alfoffo
recastsamp
le(G
ürkayn
aketal.(2007)
yieldd
ata,30m
aturities).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11
11,26
1,561,26
1,561
1,011
0,991
1,13
1,151,15
1,151,19
1,11,25*
1,11
1,141
1,151,02
CP
I6
0,98*0,98*
0,97*0,96
11
11,01
0,971
0,98*1
91,01
1,011,09
1,08**1,01
1,010,97
1,010,99
1,011,01
1,0112
1,17***1,17***
1,3*1,16*
1,05**1,05**
1,061
1,15**0,99
1,17***1,06**
11,01
1,011,66
2,27*1,66
2,27*1
0,991,01
11,01
0,583
1,021,02
1,021,02
1,011
10,85
1,020,91
1,021
PC
E6
0,970,97
0,960,95
1,011
0,970,98
0,961
0,971,01
91
1,011,01
1,011
10,96
0,940,99
0,981
112
0,980,98
0,860,96
11*
0,9*1,06*
0,981,01
0,981
11
0,991,34**
1,49***1,34**
1,49***1
1,021
0,981
0,91**3
11
0,960,98
0,931,01
0,930,99
11
11,01
PP
I6
0,980,98
0,950,97
11
0,970,99
0,970,98
0,971
91,03
1,031,03
1,041,02
1,021,05
0,971,02
1,021,03
1,0212
1,071,07
1,141,08
1,051,05
1,021
1,051
1,071,05
11,01*
1,01**1,04
1,111,04
1,111
0,991
1,091,01**
0,963
1,27*1,28
1,341,31
1,36*1,39
1,380,93**
1,260,98
1,270,82***
RD
I6
1,121,06
1,28**1,31*
1,131,23*
1,131,05
1,131,04
1,120,45***
91,04
1,011,47***
1,46***1,15***
1,27**0,21
1,021,04
1,051,04
0,3***12
0,760,75
1,64***1,54***
1,13**1,36***
0,99***0,99
0,770,98***
0,760,23***
10,99
0,981,05
0,98**1,05
0,98**1
11
0,990,99
1,083
0,94**0,93***
0,960,95
1,02***0,83
0,63**0,98
0,97*1,09*
0,95**0,89
UR
60,82***
0,81***0,85
1,060,76
0,870,84**
0,95*0,89**
0,92*0,83***
0,949
0,69***0,69***
0,790,63***
0,51*0,75**
0,64*0,94
0,91**0,95
0,7***0,8***
120,58**
0,58**0,7
0,5*0,27
0,61**0,77**
0,98**0,88**
0,93*0,59***
0,62***
11,03
1,021,07
1,061,07
1,061
0,981,02*
1,02*1,03
1,073
1,041,04
1,21,21*
1,041,35*
1,110,96
1,081,02
1,041,12
IP6
0,930,93
1,36**1,6**
1,092,09*
1,32***1
1,060,94*
0,921,09
90,82
0,841,82
1,85***1,16
3,09**1,22**
11,03
1,020,82
1,0512
0,540,61
2,942,48**
0,723,53**
0,94***1
0,651
0,550,83
Gü
rkaynak
etal.(2007)
yieldd
ata.30m
aturities,fro
m1
to30
years.MSF
Eratio
sb
etween
mo
del8
and
com
petin
gm
od
elsfo
rC
PI,P
CE
,PP
I,RD
I,UR
,and
IPfo
rall
forecastin
glead
sh
.Avalu
elow
erth
ano
ne
ind
icatesa
lower
MSF
Eo
fmo
del8
w.r.t.m
od
els1−
13.On
e,two,an
dth
reestars
mean
.10,.05,and
.01statisticalsign
ifican
ce,resp
ectively,for
the
Giaco
min
iand
Wh
ite(2006)
testw
ithq
uad
raticlo
ssfu
nctio
n.T
he
nu
mb
ero
ffo
recastsis
S ′=62
(the
first
half
of
the
S=125
com
plete
forecast
samp
le).Th
e1-step
ahead
forecasts
range
from
Septem
ber
2001to
Octo
ber
2006.Th
e12-step
ahead
forecasts
range
from
Au
gust2002
toSep
temb
er2007.T
he
forecast
perio
dco
rrespo
nd
sto
parto
fthe
Green
span
(1987-2006)an
dB
ernan
ke(2006-2012)
mo
netary
regimes.
2.8. APPENDIX 71Ta
ble
2.6.
MSF
Era
tio
sfo
rse
con
dh
alfo
ffo
reca
stsa
mp
le(G
ürk
ayn
aket
al.(
2007
)yi
eld
dat
a,30
mat
uri
ties
).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11,
040,
991
1,27
11,
271
11
11,
031,
093
1,24
1,21
1,18
1,35
1,2
1,62
0,57
*1
1,26
0,99
1,23
1,15
CP
I6
1,12
1,1
1,14
1,16
1,09
1,1
0,75
0,99
1,13
0,99
1,12
1,1
91,
041,
020,
890,
851,
01**
1,01
0,6
11*
**1
1,04
1,02
120,
99*
0,98
**0,
840,
791
10,
731,
011,
050,
990,
99*
1
11
11,
31,
071,
31,
071
0,95
10,
921
0,74
30,
970,
970,
970,
980,
92**
0,99
0,81
**0,
960,
960,
940,
970,
99P
CE
60,
980,
971
0,99
1,02
*1,
021,
020,
970,
921,
040,
981,
03*
90,
950,
96**
0,97
0,97
11
0,9*
0,99
0,88
0,95
0,95
1,01
*12
0,94
0,95
0,95
0,96
0,99
0,99
10,
990,
85*
0,94
0,94
1***
11,
051,
040,
981,
130,
981,
131
0,99
10,
991,
040,
78*
31,
121,
11,
071,
221,
081,
370,
59**
*0,
991,
090,
971,
121,
09P
PI
61,
151,
141,
13**
1,28
*1,
071,
060,
830,
981,
090,
971,
151,
079
1,01
1,04
0,97
0,99
1,01
1,01
0,68
*1,
010,
931
1,01
1,01
121,
071,
081,
031,
16**
*1,
011,
010,
780,
981,
011,
031,
071,
01
11,
021,
020,
941,
050,
941,
051
0,99
11,
041,
031,
063
1,17
1,21
1,08
1,15
0,99
1,24
0,87
1,03
1,17
1,01
1,19
1,12
RD
I6
0,86
**0,
920,
780,
780,
870,
860,
6**
10,
930,
98**
0,88
**1,
089
0,69
**0,
750,
580,
62**
*0,
75**
*0,
74**
*0,
38**
10,
7***
0,97
0,7*
*1,
03*
120,
61**
0,67
0,62
**0,
67**
0,79
***
0,75
***
0,99
***
0,99
0,64
***
0,99
0,61
**1,
06
10,
941,
111,
20,
981,
20,
981
10,
98**
*0,
91*
0,92
0,94
*3
0,92
10,
950,
720,
830,
560,
32**
*0,
960,
92*
1,02
0,88
0,6
UR
61,
021,
020,
961,
260,
940,
60,
3***
0,91
0,96
0,94
10,
669
0,98
0,97
0,91
1,06
0,87
*0,
79*
0,23
**1,
02*
0,87
0,96
0,99
0,87
121,
11**
1,1*
*0,
981,
270,
86*
1,05
**0,
43*
0,93
0,91
***
0,93
*1,
121,
16*
10,
971,
071,
130,
921,
130,
921
1,06
0,99
***
10,
960,
953
0,93
0,92
**0,
930,
850,
940,
550,
73**
0,99
0,93
0,97
0,94
0,58
IP6
0,89
0,87
0,89
1,27
0,91
0,86
0,73
1,02
0,86
*0,
980,
91,
01**
91,
17*
1,13
*1,
12**
1,54
1,06
1,4
0,77
1,02
0,98
0,96
1,18
**1,
7212
1,47
1,34
1,41
1,64
1,06
1,81
0,9
1,04
1,07
0,95
1,46
2,29
Gü
rkay
nak
etal
.(20
07)
yiel
dd
ata.
30m
atu
riti
es,f
rom
1to
30ye
ars.
MSF
Era
tio
sb
etw
een
mo
del
8an
dco
mp
etin
gm
od
els
for
CP
I,P
CE
,PP
I,R
DI,
UR
,an
dIP
for
all
fore
cast
ing
lead
sh
.Ava
lue
low
erth
ano
ne
ind
icat
esa
low
erM
SFE
ofm
od
el8
w.r
.t.m
od
els
1−1
3.O
ne,
two,
and
thre
est
ars
mea
n.1
0,.0
5,an
d.0
1st
atis
tica
lsig
nifi
can
ce,
resp
ecti
vely
,fo
rth
eG
iaco
min
ian
dW
hit
e(2
006)
test
wit
hq
uad
rati
clo
ssfu
nct
ion
.Th
en
um
ber
off
ore
cast
sis
S′ =
63(t
he
seco
nd
hal
foft
he
S=
125
com
ple
tefo
reca
stsa
mp
le).
Th
e1-
step
ahea
dfo
reca
sts
ran
gefr
om
Nov
emb
er20
06to
Jan
uar
y20
12.T
he
12-s
tep
ahea
dfo
reca
sts
ran
gefr
om
Oct
ob
er20
07to
Dec
emb
er20
12.T
he
fore
cast
per
iod
corr
esp
on
ds
top
arto
fth
eB
ern
anke
(200
6-20
12)
mo
net
ary
regi
me.
72 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVETab
le2.7.M
SFE
ratios
for
wh
ole
forecastsam
ple
(Dieb
old
and
Li(2006)yield
data,17
matu
rities).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11
0,981,12*
1,28***1,12*
1,28***1
11
11,01
0,84***3
10,99
1,01*1,01
1,021,02
0,980,99
0,990,99
1,01*1
CP
I6
0,960,94
1,061,1
1,021,02
0,98*1,02*
0,980,96
1,070,96
90,87
0,88**0,99
1,031,02
1,020,96***
1,01**0,91***
0,87*1,02
0,89***12
0,780,85**
1,041,06
1,01**1,01**
0,980,97
0,88**0,81**
1,020,85**
11,01
1,011,32*
1,69***1,32*
1,69***1
1,031
1,010,96
0,83
1,061,05
1,061,06
1,031,02
0,991,01*
1,041,06
1,011
PC
E6
0,960,96
0,970,98
11
1,011
0,940,97
0,921
90,95
0,94*0,95
0,941
10,96**
10,92**
0,950,9**
0,9912
10,97
1,03*1,02*
1,011,01
11
0,95*1
0,97***0,97*
11
0,99**1,08
1,27*1,08
1,27*1
1,131
1,011
0,873
1,031,03
1,031,03
1,011,1
10,91**
1,021,02
1,030,98
PP
I6
1,071,06
1,081,07
1,01**1,01**
11,01
1,071,07
1,081
90,97
0,960,96
11,01
1,010,99
1,010,97
0,980,98
0,9812
0,930,94
0,970,98*
0,990,99
0,991
0,94*0,94*
0,960,96
10,99
0,98*0,97
1,030,97
1,031
0,771
0,981,03
0,943
0,940,96
1,051,05
0,971,1
0,990,98
0,990,9
1,051,04
RD
I6
0,90,92
1,12**1,15*
1,11,14*
1,040,97
0,880,8
1,121
90,88**
0,91,19*
1,18*1,14
1,24*1,02
0,970,83
0,79***1,18*
1,0312
0,770,79
1,32**1,2**
1,221,41**
1,030,97
0,77*0,73**
1,29**1,02
11
11,05
1,051,05
1,051
0,99*1
0,98*0,99
1,04**3
0,930,91
1,111,05
1,021,03
0,990,84**
0,86*0,85*
0,98**0,85**
UR
60,88
0,861,1
1,041,03**
0,951
1,07*0,82
0,820,92
0,759
0,660,65
0,930,92
0,890,84
10,92
0,610,62
0,770,65
120,52
0,50,78
0,80,79
0,761,01
0,890,5
0,50,67**
0,63
11,01
1,021,08
1,11*1,08
1,11*1
0,991
11,01
1,07**3
0,980,96
1,09*1,12*
1,04*1,02
0,98*1,05
0,870,87
1,16**0,93
IP6
0,980,96
1,241,19
1,151,14
0,970,54**
0,90,88
1,231,06
90,84
0,821
1,131,06
1,020,95
1,070,79
0,791,08
1,0312
0,62**0,64***
0,670,65
0,990,87
0,95***1,02
0,63**0,63***
0,80,96
Dieb
oldan
dLi(2006)yield
data.17
matu
ritiescorresp
ond
ing
to3,6,9,12,15,18,21,24,30,36,48,60,72,84,96,108,an
d120
mon
ths.M
SFE
ratiosb
etween
mod
el8an
dm
od
els1−
13fo
rC
PI,P
CE
,PP
I,RD
I,UR
,and
IPfo
rallfo
recasting
leads
h.A
value
lower
than
on
ein
dicates
alow
erM
SFE
ofm
od
el8w
.r.t.the
com
petin
gm
od
els.On
e,tw
o,and
three
starsm
ean.10,.05,an
d.01
statisticalsignifi
cance,resp
ectively,for
the
Giaco
min
iand
Wh
ite(2006)
testw
ithq
uad
raticlo
ssfu
nctio
n.T
he
nu
mb
ero
fforecasts
isS=
150.Th
e1-step
ahead
forecastsran
gefrom
July
1987to
Decem
ber
1999.Th
e12-step
ahead
forecastsran
gefrom
Jun
e1988
toN
ovemb
er2000.T
he
forecastp
eriod
corresp
on
ds
top
artofth
eG
reensp
an(1987-2006)
mo
netary
regimes.
2.8. APPENDIX 73Ta
ble
2.8.
MSF
Era
tio
sfo
rfi
rsth
alfo
ffo
reca
stsa
mp
le(D
ieb
old
and
Li(2
006)
yiel
dd
ata,
17m
atu
riti
es).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11
1*1,
071,
21*
1,07
1,21
*1
11
11
0,84
**3
1,02
1,03
1,05
1,06
1,05
1,08
0,99
11,
011,
021,
031,
02C
PI
60,
950,
971,
091,
131,
021,
020,
981,
010,
990,
961,
080,
999
0,81
**0,
89**
0,95
1,01
1,02
1,02
0,94
**1
0,89
*0,
82**
10,
9512
0,71
**0,
87**
1,03
1,03
0,99
***
0,99
***
0,97
0,97
0,89
***
0,76
**0,
990,
93**
*
11,
011,
011,
311,
69**
1,31
1,69
**1
1,01
11,
010,
990,
843
1,07
1,06
1,07
1,08
1,05
1,02
0,99
1*1,
051,
071,
051*
PC
E6
0,96
0,95
0,97
0,98
11
0,99
10,
940,
960,
941
90,
94*
0,93
**0,
950,
941
10,
981
0,91
**0,
940,
930,
9912
0,99
*0,
96*
1,02
1,02
1,01
1,01
11,
01*
0,93
***
0,99
**0,
99*
0,96
**
11
11,
061,
261,
061,
261
1,14
11,
010,
990,
883
1,04
1,04
1,04
1,04
1,01
1,15
1*0,
93*
1,02
1,03
1,03
0,99
PP
I6
1,09
1,09
1,1
1,08
1,01
*1,
01*
1,01
1,01
1,09
1,1
1,08
1,02
***
90,
97*
0,98
0,94
0,99
11
0,99
*1,
010,
990,
980,
971,
0112
0,91
**0,
950,
950,
95**
0,98
0,98
11
0,95
0,94
**0,
931
10,
991
0,92
0,99
0,92
0,99
10,
631
0,97
10,
963
11,
020,
960,
970,
861,
030,
950,
991,
010,
890,
971,
11R
DI
60,
920,
910,
960,
990,
971,
010,
970,
990,
840,
730,
961,
059
0,98
0,95
1,02
1,01
0,98
1,08
0,91
***
1,01
0,8
0,76
0,99
1,12
120,
90,
881,
060,
910,
951,
15*
0,92
***
1,1
0,74
0,7
1,01
1,1
11,
03**
1,02
1,12
1,13
1,12
1,13
11,
011*
*0,
981,
071,
13**
*3
0,85
***
0,82
***
1,22
***
1,16
***
1,06
0,94
1,03
0,75
**0,
74**
0,75
**1,
040,
63**
UR
60,
780,
771,
2**
1,11
**1,
1***
1,01
1,04
1,17
**0,
710,
72*
1,05
**0,
69
0,55
*0,
54*
0,93
0,96
0,88
0,89
1,05
**0,
99**
0,51
*0,
52*
0,86
0,51
120,
410,
40,
730,
74**
0,73
0,76
1,07
0,8*
0,4
0,4*
0,69
**0,
49
11,
04*
1,04
*1,
091,
121,
091,
121
1,01
1**
11,
121,
053
0,91
0,89
*1,
2**
1,3*
**1,
25**
*1,
230,
961,
070,
77**
0,78
1,35
***
0,85
IP6
0,96
0,94
1,59
***
1,57
***
1,38
**1,
99**
0,95
**0,
53*
0,85
0,85
1,71
***
1,06
90,
84*
0,83
**1,
54**
1,6*
*1,
32**
2,09
**0,
92*
1,11
0,76
*0,
781,
82**
*0,
99**
120,
65**
0,71
**1,
261,
24*
1,43
1,89
**0,
91**
0,97
0,63
**0,
65**
1,52
0,92
**
Die
bo
ldan
dL
i(20
06)
yiel
dd
ata.
17m
atu
riti
esco
rres
po
nd
ing
to3,
6,9,
12,1
5,18
,21,
24,3
0,36
,48,
60,7
2,84
,96,
108,
and
120
mo
nth
s.M
SFE
rati
os
bet
wee
nm
od
el8
and
mo
del
s1−1
3fo
rC
PI,
PC
E,P
PI,
RD
I,U
R,a
nd
IPfo
ral
lfo
reca
stin
gle
ads
h.A
valu
elo
wer
than
on
ein
dic
ates
alo
wer
MSF
Eo
fmo
del
8w
.r.t
.th
eco
mp
etin
gm
od
els.
On
e,tw
o,an
dth
ree
star
sm
ean
.10,
.05,
and
.01
stat
isti
cals
ign
ifica
nce
,res
pec
tive
ly,f
or
the
Gia
com
inia
nd
Wh
ite
(200
6)te
stw
ith
qu
adra
tic
loss
fun
ctio
n.T
he
nu
mb
ero
ffo
reca
sts
isS′ =
75(t
he
firs
th
alfo
fth
eS=
150
com
ple
tefo
reca
stsa
mp
le).
Th
e1-
step
ahea
dfo
reca
sts
ran
gefr
om
July
1987
toSe
pte
mb
er19
93.T
he
12-s
tep
ahea
dfo
reca
sts
ran
gefr
om
Jun
e19
88to
Au
gust
1994
.Th
efo
reca
stp
erio
dco
rres
po
nd
sto
par
toft
he
Gre
ensp
an(1
987-
2006
)m
on
etar
yre
gim
e.
74 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVETab
le2.9.M
SFE
ratios
for
secon
dh
alfoffo
recastsamp
le(D
iebo
ldan
dLi(2006)
yieldd
ata,17m
aturities).
hm
od1
mod
2m
od3
mod
4m
od5
mod
6m
od7
mod
9m
od10
mod
11m
od12
mod
13
11,01
0,961,2
1,37**1,2
1,37**1
11
11,02
0,85**3
0,960,95*
0,970,95
0,970,94
0,960,98*
0,97*0,96
0,980,96
CP
I6
10,89
1,011,03
1,021,02
1*1,03
0,970,98
1,05**0,9
91
0,871,06**
1,05**1,02
1,020,98*
1,02***0,93*
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1,081,13
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3FORECASTING THE GLOBAL MEAN SEA LEVEL
A CONTINUOUS-TIME STATE-SPACE APPROACH
Lorenzo Boldrini
Aarhus University and CREATES
75
76 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Abstract
In this paper we propose a continuous-time, Gaussian, linear, state-space system
to model the relation between global mean sea level (GMSL) and the global mean
temperature (GMT), with the aim of making long-term projections for the GMSL. We
provide a justification for the model specification based on popular semi-empirical
methods present in the literature and on zero-dimensional energy balance models.
We show that some of the models developed in the literature on semi-empirical
models can be analysed within this framework. We use the sea-level data recon-
struction developed in Church and White (2011) and the temperature reconstruction
from Hansen et al. (2010). We compare the forecasting performance of the proposed
specification to the procedures developed in Rahmstorf (2007b) and Vermeer and
Rahmstorf (2009). Finally, we compute projections for the sea-level rise conditional
on the 21st century SRES temperature scenarios of the IPCC fourth assessment report.
Furthermore, we propose a bootstrap procedure to compute confidence intervals for
the projections, based on the method introduced in Rodriguez and Ruiz (2009).
3.1. INTRODUCTION 77
3.1 Introduction
Climate changes, the increase in the global temperature and sea level are long-
standing topics. Monitoring and predicting the rise in the sea level is of great impor-
tance due to its close relation with global climate changes and the socio-economic
effects it entails. In particular, the sea-level rise has direct consequences for popu-
lations living near the current mean sea level, Anthoff, Nicholls, Tol, and Vafeidis
(2006), Anthoff, Nicholls, and Tol (2010), Arnell, Tompkins, Adger, and Delaney (2005),
Sugiyama, Nicholls, and Vafeidis (2008). Physical and statistical models are needed
to measure the rate of change of the sea level and understand its relation to anthro-
pogenic and natural causes.
In this paper we propose a statistical framework to model the relation between
the global mean sea level (GMSL) and the global mean temperature (GMT), with
the aim of making long-term projections for the GMSL. The model belongs to the
class of semi-empirical models. We provide a justification for the model specifi-
cation based on popular semi-empirical methods present in the literature and on
zero-dimensional energy balance models. We show that some of the semi-empirical
models developed in the literature to study the relation between sea-level rise and
temperature can be analysed within this framework.
To date, there are two methods of estimating the sea-level rise as a function of
climate forcing. The conventional approach, used by the Intergovernmental Panel
on Climate Change (IPCC) climate assessments, is to use process-based models to
estimate contributions from the sea-level components and then sum them to obtain
an estimate of the sea-level increase, see for instance Meehl, Covey, Taylor, Delworth,
Stouffer, Latif, McAvaney, and Mitchell (2007a), Meehl, Stocker, Collins, Friedlingstein,
Gaye, Gregory, Kitoh, Knutti, Murphy, Noda, et al. (2007b), Pardaens, Lowe, Brown,
Nicholls, and De Gusmão (2011), Solomon, Plattner, Knutti, and Friedlingstein (2009).
Variations in the sea level originate from steric, eustatic, and non-climate changes.
By steric, we mean sea-level variations due to ocean volume changes, resulting from
temperature (thermosteric) and salinity (halosteric) variations. By eustatic, we mean
variations in the mass of the oceans as a result of water exchanges between the oceans
and other surface reservoirs (ice sheets, glaciers, land water reservoirs, and the at-
mosphere)1. By non-climate causes we mean variations in the quantity of water in
the oceans due to human impact, such as the building of dams and the extraction of
groundwater. However, the theoretical understanding of the different contributors is
incomplete, as IPCC models under-predict rates of sea-level increase.
The alternative way of making projections of the sea level is the class of semi-
empirical models. These models analyse statistical relationships using physically
plausible models of reduced complexity in which the sea-level rate of change de-
pends on the global temperature. The main idea behind semi-empirical models is
1In this paper we adopt the same definitions of steric and eustatic used in Cazenave and Nerem (2004).
78 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
that the steric and eustatic contributors to the sea level (the major ones) respond
to changes in the global temperature. The first semi-empirical model was proposed
by Gornitz, Lebedeff, and Hansen (1982) who specify a linear relation between sea
level and temperature. Some more recent models where developed by specifying
a differential equation, relating the sea level to temperature or other climate forc-
ing. Representative examples are Rahmstorf (2007b), Vermeer and Rahmstorf (2009),
Grinsted, Moore, and Jevrejeva (2010), Kemp, Horton, Donnelly, Mann, Vermeer, and
Rahmstorf (2011), Jevrejeva, Grinsted, and Moore (2009), Jevrejeva, Moore, and Grin-
sted (2010), Jevrejeva, Moore, and Grinsted (2012b), Jevrejeva, Moore, and Grinsted
(2012a).
All semi-empirical models project higher sea-level rise, for the 21st century, that
the last generation of process-based models, summarized in the IPCC Fourth As-
sessment Report, see for instance Moore, Grinsted, Zwinger, and Jevrejeva (2013),
Cazenave and Nerem (2004), Munk (2002), and Rahmstorf (2007b). A comprehensive
survey on the different process-based and semi-empirical models can be found in
Moore et al. (2013).
We propose a state-space approach to forecast the global mean sea level, condi-
tional on the global mean temperature. State-space systems allow to address the prob-
lems of smoothing, detrending, and parameter estimation in a unique framework.
We consider in particular continuous-time, linear, Gaussian state-space systems of
the type described in Bergstrom (1997). More specifically, the state vector follows
a multivariate, Gaussian, Ornstein–Uhlenbeck process. The discretised system pre-
serves its linearity and Kalman filtering techniques apply. In particular, the Kalman
filter is used for two tasks: the first one is to compute the likelihood function of the
state-space system, needed for parameter estimation, and the second one is to make
forecasts of the sea level, conditional on the temperature.
The statistical framework of state-space systems allows to distinguish between
measurement noise and model uncertainty, through the measurement and state
equations, respectively. Furthermore, in this setup it is possible to consider different
levels of measurement noise for different points in time. In fact, the reconstructions
of the sea level, in particular, are typically very noisy and the measurements uncer-
tainty reflects, for instance, the changes in the measurement instruments throughout
the decades, as well as the changes in the data sources. In this study we use the
sea-level reconstruction from Church and White (2011), who also provide an estimate
of the uncertainty of their sea-level estimates. In the state-space approach these
uncertainties (measured as standard deviations) are directly used to parameterise
the time-varying variances of the measurement errors of the sea-level time series.
Temperature data are taken from Hansen et al. (2010). Both sea level and temperature
data correspond to monthly reconstructions.
We provide a justification for the model specification based on popular semi-
empirical methods present in the literature and on zero-dimensional energy balance
3.2. MODEL SPECIFICATION 79
models. In more detail, we specified the system dynamics of sea level and tempera-
ture as well as the functional form linking these variables, consistently with the ones
suggested by the existing literature. We show that some of the semi-empirical models
developed in the literature to study the relation between sea level and temperature,
can be analysed within this framework.
Semi-empirical models are usually specified as differential equations, as in Rahm-
storf (2007b), Vermeer and Rahmstorf (2009), Grinsted et al. (2010), Kemp et al. (2011),
Jevrejeva et al. (2009), Jevrejeva et al. (2010), Jevrejeva et al. (2012b), and Jevrejeva et al.
(2012a). By specifying the state-space system in continuous time and then deriving
the exact discrete-time system, we can make inference on the structural parameters
driving the continuous-time process. A similar approach was used in Pretis (2015)
(forthcoming), in which the author shows the equivalence of a two-component en-
ergy balance model to a cointegrated system. He then shows the exact mapping
between the continuous-time system to the discrete-time one, that amounts to a
cointegrated vector autoregressive system.
In the literature on semi-empirical models, state-space system representations
and the Kalman filter are often used with the aim of assimilating noisy measure-
ments from different sources2. Such studies are for instance, Miller and Cane (1989)
and Chan, Kadane, Miller, and Palma (1996) who use the Kalman filter with an un-
derlying physical model to assimilate average sea-level anomalies from tide gauge
measurements and Cane, Kaplan, Miller, Tang, Hackert, and Busalacchi (1996) and
Hay, Morrow, Kopp, and Mitrovica (2013).
The paper is organised as follows: in Section 3.2, we explain the statistical frame-
work and introduce the model specification, showing how it relates to some impor-
tant models in the literature; in Section 3.3, we describe the dataset; in Section 3.4,
we illustrate the forecasting procedures; in Section 3.5, we provide some details on
the computational aspects of the analysis; in Section 3.6, we present results; finally,
Section 3.7 concludes.
3.2 Model specification
Energy balance models and temperature dynamics
In this section we present the foundations for the temperature process used in the
state-space model proposed in Section 3.2. This section draws heavily on North,
Cahalan, and Coakley (1981) and Imkeller (2001) and we refer the reader to these
papers for more details. The starting model, for the temperature, belongs to the class
of zero-dimensional energy balance models (EBMs), detailed in North et al. (1981),
2In this context, assimilation usually refers to the filtering of a variable, measured at a specific pointin time and geographic location, by taking into account information from measurements taken at neigh-bouring locations at the same point in time. If the filtered variables are output from the Kalman filter, thecontribution of variable j to the estimate of variable i is given by element (i , j ) of the Kalman gain matrix.
80 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
and whose review is based on the models introduced in Budyko (1968), Budyko
(1969), Budyko (1972), and Sellers (1969). These models are based on thermodynamic
concepts and global radiative heat balance, for the Earth system. This type of models
describe the global temperature process as (possibly stochastic) univariate, differen-
tial equations. In particular, the change in the Planet’s global temperature at time t is
seen as a function of the difference between the incoming and outgoing radiation.
The incoming (absorbed) radiation Ri n is caused by solar irradiance (i.e. the
sunlight reaching the Earth) and is affected by the reflectivity of the Planet. The in-
coming radiation is then a function of the solar constant3 σ0, the albedo coefficient4
α, and the radius R of the Earth, in particular: Ri n = σ0(1−α)πR2. The outgoing
radiation Rout is assumed, for simplicity, to be black-body5 radiation, obeying the
Stefan-Boltzmann law6. It is then a function of the absolute temperature T of the
Planet and its radius R, in particular: Rout = 4πR2kSB T 4.
The analysis of zero-dimensional EBMs begins with the concept of global ra-
diative heat balance. In radiative equilibrium the rate at which solar radiation is
absorbed by the Planet matches the rate at which infrared radiation is emitted by it.
The condition of radiative equilibrium is given by
Rout︷ ︸︸ ︷4πR2kSB T 4 =
Ri n︷ ︸︸ ︷σ0(1−α)πR2, (3.1)
where T is the effective radiating temperature of the Planet, and kSB = 0.56687 ·10−7[W m−2K 4], is the Stefan-Boltzmann constant. Note that both sides of equation
(3.1) are expressed in units of power, in particular in Watts [W ]. When the incoming ra-
diation does not match the outgoing radiation, the temperature of the Planet changes
in order to compensate the disequilibrium. The time-evolution of the temperature
can then be modelled with the following zero-dimensional EBM:
CdT (t )
d t= Ri n −Rout
= σ0(1−α)πR2 −4πR2kSB T 4(t ), (3.2)
where C , that has units of[
W ·sK
]=
[JK
], represents global thermal inertia and regulates
the speed of the temperature response. With T (t ) we make explicit the dependence
3The solar constant is a measure of the mean solar electromagnetic radiation per unit area that wouldbe incident on a plane perpendicular to the sun rays.
4The term albedo, Latin for white, describes the average reflection coefficient of an object. Thegreenhouse effect, for instance, can lower the albedo of the Earth, and cause global warming.
5A black body is an idealized physical object that absorbs all incident electromagnetic radiation. Thetotal energy per unit of time, per unit of surface area, radiated by a black body depends solely on itsabsolute temperature and obeys the Stefan-Boltzman law.
6The Stefan-Boltzmann law describes the power radiated from a black body in terms of its thermody-namic temperature. The thermodynamic temperature (absolute temperature) is commonly expressed inKelvin [K ], where 0[K ] =−273.15[°C] corresponds to the lowest achievable temperature, according to thethird principle of thermodynamics.
3.2. MODEL SPECIFICATION 81
of temperature on time. Equation (3.2) can be written as
CdT (t )
d t= Qα−γT 4(t ),
(3.3)
where Q is a constant proportional to σ0, γ is a constant proportional to kSB , and
α= (1−α) is the co-albedo. Note that equation (3.3) allows to relax the black-body
assumption. In fact, for a so called grey body7 we have that the emissive power, per
unit surface area is I = εkSB T 4, with ε< 1.
Equation (3.3) is purely deterministic. To allow for random forcing, stochastic
EBM have been introduced, see for instance Fraedrich (1978) and Hasselmann (1976).
A stochastic EBM can be written in the following way:
CdT (t )
d t= Qα−γT 4(t )+W (t ), (3.4)
where W (t ) is a white noise random forcing8.
Depending on the time scale under examination, the solar constant funtion Q
can be allowed to be time-varying. For instance, the Milankovich cycle responsible for
the glaciations, i.e. a cyclical mutation in the eccentricity of the Earth orbit due to the
gravitational pull of other planets, has a period of approximately 105 years and can be
expressed as Q(t ) =Q0+si n(ωt ) withω= 10−5[1/year ], see Imkeller (2001). Similarly,
the co-albedo α can be assumed to be time-varying and, in particular, to depend on
the global temperature, i.e. α(T (t )). This is motivated, among other reasons, by the
ice-cap feedback. That is, the albedo of the Planet changes with the temperature as a
result of the shrinking or spreading of ice sheets on the Earth’s surface, that depends
on the global temperature.
Taking into consideration these arguments, equation (3.4) can be written in the
form
CdT (t )
d t= Ri n −Rout +W (t )
= Q(t )α(T (t ))−γT 4(t )+W (t ). (3.5)
Different specifications for Ri n and Rout have been suggested in the literature. Budyko
(1969), for instance, suggested that the infrared radiation to space, Rout , can be
represented as a linear function of the surface temperature T , that is:
4πR2σT 4(t ) ∼= 4πR2σ(δ1 +δ2T (t )), (3.6)
where δ1, and δ2 are constants, taking into account factors such as average cloudiness,
the effects of infrared absorbing gases, and the variability of water vapor. Sellers (1969)
7A body that does not absorb all incident radiation.8The white noise process is defined as E [W (t)] and E [W (t)W (t ′)] = q2δ(t − t ′) where δ(t − t ′) is a
Dirac delta, and q is a constant, see also Nicolis (1982).
82 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
suggested taking a linear approximation also of the albedo, or similarly, of the co-
albedo:
α(T (t )) =β1 +β2T (t ), (3.7)
where β1 and β2 are constants.
We now show that some important special cases of model (3.5) can be written in
the form
dT (t ) =
bT +κTµT (t )+aT S S(t )+aT T T (t )
d t +dηT (t ), (3.8)
where bT , aT S , aT T , and κT are constants, µT (t) is a time-varying process, S(t) is
the sea-level process, and ηT (t ) is a scaled Brownian motion with E [dηT (t )dηT (t )] =ΣT T d t . In particular, equation (3.8) is an exact representation of model (3.5) for these
specifications:
I. for a time-invariant solar constant Q, a constant co-albedo α, and taking a
linear approximation to Rout = γT 4(t) ∼= δ1 +δ2T (t), we obtain the following
relation between the components of equation (3.5) and the ones of equation
(3.8):
bT = 1
C(Qα−δ1), κTµT (t ) = 0, aT S = 0, aT T =− 1
Cδ2, dηT (t ) 6= 0;
(3.9)
II. for a time-varying solar constant Q(t), a constant co-albedo α, and taking a
linear approximation to Rout = γT 4(t ) ∼= δ1 +δ2T (t ), we obtain:
bT =− 1
Cδ1, κTµT (t ) = 1
CQ(t )α, aT S = 0, aT T =− 1
Cδ2, dηT (t ) 6= 0;
(3.10)
III. for a time-varying solar constant Q(t), a time varying co-albedo α(t), and a
linear approximation to Rout = γT 4(t ) ∼= δ1 +δ2T (t ), we have:
bT =− 1
Cδ1, κTµT (t ) = 1
CQ(t )α(t ), aT S = 0, aT T =− 1
Cδ2, dηT (t ) 6= 0;
(3.11)
IV. for a time-invariant solar constant Q, a time varying co-albedo, assuming
dependence on the temperature of the co-albedo α(T (t )) and linearity in the
temperature α(T (t )) =β1 +β2T (t ), and taking a linear approximation Rout =γT 4(t ) ∼= δ1 +δ2T (t ), we obtain:
bT = 1
C(Qβ1 −δ1), κTµT (t ) = 0, aT S = 0, aT T = 1
C(Qβ2 −δ2), dηT (t ) 6= 0.
(3.12)
3.2. MODEL SPECIFICATION 83
The time-varying component κTµT (t) controls for changes in the albedo and/or
solar constant coefficients. That is, it accounts among other things, for the effects
of greenhouse gases and other phenomena that change the radiation-absorbing
properties of the Planet.
Semi-empirical models and sea-level dynamics
Some of the most representative semi-empirical models in the literature are here
briefly presented. Throughout this section we denote with t time, S(t) the global
mean sea level, and with T (t ) the global mean temperature. Parameters are indicated
with lower case letters. We consider the following five models:
I. Gornitz et al. (1982) suggest the following link between sea level and tempera-
ture:
S∗(t ) = aT ∗ (t − t0
)+b, (3.13)
where S∗ and T ∗ are the 5-year averages of the global sea level and temperature,
respectively. The parameters a and b are estimated by least-squares linear
regression and the time lag t0 is chosen to minimize the variance between
(3.13) and the sea-level curve.
II. Rahmstorf (2007b) suggests the following differential equation relating sea level
to temperature:
dS(t )
d t= r
(T (t )−T0
), (3.14)
where r is a parameter to be estimated. The sea-level rise above the previous
equilibrium state can be computed as
S(t ) = r∫ t
t0
(T (s)−T0
)d s. (3.15)
The statistical analysis in Rahmstorf (2007b) is comprised of several steps. First,
the GMSL and GMT series are processed to obtain annual means. Second, a
singular spectrum analysis filter, with a 15-year smoothing period, is applied to
the series of yearly averages. Third, data is divided into 5 years bins, in which
the average is taken. Lastly, the resulting sea-level series in first differences is
regressed on the resulting temperature in levels (with optional detrending of
both series before the regression). The data they use are the global mean sea
level from Church and White (2006) and the global temperature anomalies data
from GISTemp, Hansen, Ruedy, Sato, Imhoff, Lawrence, Easterling, Peterson,
and Karl (2001). See Holgate, Jevrejeva, Woodworth, and Brewer (2007) and
Rahmstorf (2007a) for comments on the statistical procedure.
84 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
III. Vermeer and Rahmstorf (2009) suggest the following extension of the previous
model:
dS(t )
d t= v1
(T (t )−T0
)+ v2dT (t )
d t. (3.16)
In this model the authors add the term v2dT (t )
d t to the Rahmstorf (2007b) model,
corresponding to an “instantaneous” sea-level response. The statistical method-
ology is similar to the one in Rahmstorf (2007b) and a thorough description of
it can be found in the online appendix to their paper.
IV. Kemp et al. (2011) propose the following model:
dS(t )
d t= k1
(T (t )−T0,0
)+k2(T (t )−T0(t )
)+k3dT (t )
d t,
dT0(t )
d t= T (t )−T0(t )
τ. (3.17)
The first term captures a slow response compared to the time scale of interest,
the second one captures intermediate time scales, where an initial linear rise
gradually saturates with time scale τ as the base temperature T0 catches up with
T (t). In Rahmstorf (2007b), equation (3.14), T0 was assumed to be constant.
The third term is the immediate response term introduced by Vermeer and
Rahmstorf (2009).
V. Grinsted et al. (2010) propose the following model:
Seq = g1T + g2,
dS(t )
d t= Seq −S(t )
τ, (3.18)
where Seq is the equilibrium sea level, for a given temperature. They assume a
linear approximation of the relation between sea level and temperature, due to
the closeness of the current sea level to the equilibrium in this climate period
(late Holocene-Anthropocene) and for small changes in the sea level. Equation
(3.18) can be integrated to give the sea level S over time, using the history of the
temperature T and knowledge of the initial sea level at the start of integration
S0. They impose constraints on the model, suggested by reasonable physical
assumptions.
We here show how some of the different model specifications for the relation between
sea level and temperature can be seen as special cases of the following stochastic
differential equation
dS(t ) =
bS +κSµS (t )+aSS S(t )+aST T (t )
d t +dηS (t ), (3.19)
3.2. MODEL SPECIFICATION 85
where bS , κS , aSS , and aST are parameters, µS (t) is a time-dependent process, and
ηS (t ) is a scaled Brownian motion with E [dηS (t )dηS (t )] =ΣSS d t . In particular, mod-
els II and V can be written as particular cases of model (3.19) if some restrictions are
imposed on its components:
II. for the Rahmstorf (2007b) specification we have the following relation between
the components of equation (3.19) and equation (3.14):
bS =−r T0, κSµS (t ) = 0, aSS = 0, aST = r, dηS (t ) = 0; (3.20)
V. for the Grinsted et al. (2010) specification, the relation between the components
of equation (3.19) and (3.18) are:
bS = 1
τg2, κSµS (t ) = 0, aSS =−1
τ, aST = 1
τg1, dηS (t ) = 0. (3.21)
Note that in models II and V the component κSµS (t) is zero. This reflects the fact
that in both papers the series of temperature and sea level are detrended before the
parameter estimation. Instead, we prefer to model the trend component jointly with
the temperature and sea-level dynamics.
State-space system
Combining the temperature and sea-level dynamics, equations (3.8) and (3.19), we
obtain the following multivariate process
d
[S(t )
T (t )
]= bd t +
[aSS aST
aT S aT T
][S(t )
T (t )
]d t +
[κS 0
0 κT
][µS (t )
µT (t )
]d t +dη(t ),
(3.22)
where b = [bS : bT ]′, µ(t) = [µS (t) : µT (t)]′ is the trend component, and dη(t) =[dηS (t) : dηT (t)]′ where η(t) is a scaled, 2-dimensional, Brownian motion with
E [dη(t )dη(t )′] =Σd t and
Σ =[ΣSS ΣST
ΣT S ΣT T
], (3.23)
a positive semidefinite covariance matrix.
We consider two parametric forms for the trend componentµ(t ), namely a linear and
a quadratic trend:
(i) linear trend component,
dµ(t ) = λl d t , (3.24)
where λl = [λSl
:λTl ] are parameters to be estimated;
86 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
(ii) quadratic trend component,
dµ(t ) = λ(t )d t ,
dλ(t ) = νq d t , (3.25)
where λ(t ) = [λS (t ) :λT (t )]′ is a 2-dimensional process and νq = [νSq : νT
q ]′ are
parameters to be estimated.
The choice of the trend components was driven by the forecast performance of the
models, according to the forecasting exercise detailed in Section 3.4.
State equation
We now show how to obtain the exact discrete representation of the continuous-time
state-space system (3.22) with linear trend (3.24) (the derivation for the model with
the quadratic trend (3.25) is analogous). The system of equations (3.22)-(3.24) can
be written in compact form, delivering the following Gaussian, Ornstein-Uhlenbeck
(OU) process:
dα(t ) = cd t +Aα(t )d t +dξ(t ), (3.26)
where α(t ) = [S(t ),T (t ),µS (t ),µT (t )]′, µS (t ) and µT (t ) indicate the trends for the sea-
level and the temperature, respectively, dξ(t) = [dη(t)′ : 0′]′, c = [b′ :λ′l ]′ = [0′ :λ′
l ]′,and the autoregressive matrix has the form
A =
aSS aST 1 0
aT S aT T 0 1
0 0 0 0
0 0 0 0
, (3.27)
where we constrain the two parameters κS = κT = 1, as suggested in Bergstrom
(1997), in order to avoid identification issues. We set the intercept b = 0 because of
the presence of trend component. The exact discrete state-space representation can
be recovered from the continuous-time equations following, for instance, Bergstrom
(1997). The solution of the OU process (3.22) is
α(t ) = e tAα(0)+∫ t
0eA(t−s)cd s +
∫ t
0eA(t−s)dξ(s), (3.28)
whereα(0) is the initial value of the system. Note that the solution (3.28) always exists.
Denote with τ= 1, . . . ,n, n ∈N, the time instances at which the sea-level and the
temperature processes are sampled (measured), i.e. α(t = τ) =ατ. The relationship
between the state vector at time τ and time τ+1 derives from equation (3.28) and is
given by
3.2. MODEL SPECIFICATION 87
ατ+1 = eAατ+∫ τ+1
τeA(τ+1−s)cd s +
∫ τ+1
τeA(τ+1−s)dξ(s). (3.29)
Equation (3.29) corresponds to a Gaussian, vector autoregressive process of this form
ατ+1 = c∗+A∗ατ+ξτ, (3.30)
with ξτ ∼ N (0,Σ∗), where
c∗ =∫ 1
0eA(1−s)cd s,
A∗ = eA,
Σ∗ =∫ 1
0eA(1−s)ΣeA′(1−s)d s. (3.31)
The constants aSS , aST , aT S , aT T , λ, and Σ are parameters to be estimated.
Measurement equation
The global mean sea level and the global mean temperature can be seen as stock vari-
ables, sampled at time instances τ and subject to measurement error, see for instance
Harvey and Stock (1993). Let Srτ and T r
τ be the reconstructed (or measured) sea-level
and temperature processes, respectively, and Sτ and Tτ the true (unobserved), latent
ones. The measurement equation for system (3.30) is thus:[Srτ
T rτ
]=
[SτTτ
]+
[εSτ
εTτ
], (3.32)
where ετ = [εSτ : εT
τ ]′ ∼ N(0,Hτ
)is a bivariate random vector of independent mea-
surement errors. Note that the variance-covariance matrix is allowed to vary through
time, in particular
Hτ =[σ2,Sτ 0
0 σ2,T
]. (3.33)
The variance of the measurement error for the sea level σ2,Sτ , is allowed to change
in time. In particular, in this work we use the sea-level reconstruction from Church
and White (2011). In their analysis, the authors provide uncertainty estimates of
the sea-level reconstruction at each point in time. The change in the uncertainty of
the reconstructed sea-level series reflects the change in time of the measurement
instruments, as well as the change in the data sources. Note that in this context, the
magnitude of the observation error variances controls the smoothness of the filtered
series of sea level and temperature. The parameterσ2,T is estimated together with the
other system parameters, whereas the sequence σ2,Sτ τ=1:n is fixed and corresponds
88 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
to the uncertainty values reported in Church and White (2011).
Combining equations (3.30) with (3.32) we obtain a linear, Gaussian, state-space
system (see for instance Brockwell and Davis (2009) and Durbin and Koopman
(2012)):
[Srτ
T rτ
]=
[1 0 0 0
0 1 0 0
]SτTτµSτ
µTτ
+ετ, ετ ∼ N(0,Hτ
),
Sτ+1
Tτ+1
µSτ+1
µTτ+1
= c∗+A∗
SτTτµSτ
µTτ
+ξτ, ξτ ∼ N(0,Σ∗)
. (3.34)
System (3.34) is linear in the state variables and Kalman filtering/smoothing tech-
niques apply, allowing to estimate the system parameters by maximum likelihood,
see for instance Durbin and Koopman (2012).
In the semi-empirical literature, dynamic models of sea level and temperature
are usually formulated in continuous time. A clear mapping between a multivari-
ate, Gaussian, Ornstein-Uhlenbeck process and its discrete-time analogue allows to
make inference on the parameters of the original process, introducing no bias due
to discretizations. A convenient aspect of specifying the model in state-space form
is that measurement noise and trends can be modelled in a joint framework. In this
way the problem of smoothing, detrending, and parameter inference can be handled
in a unified framework.
The dimensional analysis for the continuous-time and discrete-time systems is
provided in Appendix 3.9.
3.3 Data
• Temperatures. The temperature data are taken from the GISS dataset, Com-
bined Land-Surface Air and Sea-Surface Water Temperature Anomalies (Land-
Ocean Temperature Index, LOTI). The values are temperature anomalies, i.e.
deviations from the corresponding 1951-1980 means, Hansen et al. (2010)9.
The time series we use is composed of mean global monthly values. The values
of the original series are in centi-degrees Celsius ([c°C] = 10−2[°C]).
• Sea level. The sea-level data is from Church and White (2011)10. The authors
also provide uncertainty estimates for each measurement. They estimate the
rise in global average sea level from satellite altimeter data for 1993-2009 and
9Data can be downloaded from http://data.giss.nasa.gov/gistemp/.10Data can be downloaded from http://www.cmar.csiro.au/sealevel/sl_data_cmar.html.
3.3. DATA 89
from coastal and island sea-level measurements from 1880 to 2009. The mea-
surements of the original series are in millimetres [mm].
In our analysis we use monthly observations ranging from January 1880 to December
2009, range in which the two series overlap, for a sample size equal to 1560.
• IPCC temperature scenarios. These series correspond to reconstructed and
simulated annual temperatures, from 1900 to 2099 from the 2007 IPCC Fourth
Assessment Report, SRES scenarios. In particular, we use the A1b, A2, B1, and
commit groups of temperature scenarios11. The 4 groups correspond to dif-
ferent “storylines”12. The storylines “describe the relationships between the
forces driving greenhouse gas and aerosol emissions and their evolution during
the 21st century for large world regions and globally. Each storyline represents
different demographic, social, economic, technological, and environmental
developments that diverge in increasingly irreversible ways.” (Carter (2007,
page 9)). Each group has a different number of scenarios and in total there are
75 scenarios. The 4 scenarios groups can be described in the following way:
(i) A1b group. This group belongs to the A1 storyline and scenario family,
that is “a future World of very rapid economic growth, global population
that peaks in mid-century and declines thereafter, and rapid introduc-
tion of new and more efficient technologies.” (Carter (2007, page 9)) in
which an intermediate level of emissions has been assumed. There are 21
scenarios belonging to this group.
(ii) A2 group. “A very heterogeneous World with continuously increasing
global population and regionally oriented economic growth that is more
fragmented and slower than in other storylines.” (Carter (2007, page 9)).
There are 17 scenarios belonging to this group.
(iii) B1 group. “A convergent World with the same global population as in
the A1 storyline but with rapid changes in economic structures toward a
service and information economy, with reductions in materials intensity,
and the introduction of clean and resource-efficient technologies.” (Carter
(2007, page 9)). There are 21 scenarios belonging to this group.
(iv) commit group. In this group of scenarios, the World’s countries commit
to lower greenhouse gases emissions. There are 16 scenarios belonging to
this group.
The A1b and A2 groups reflect scenarios with an acceleration in temperature
growth (high temperature increase); the B1 group reflects scenarios of constant
11Data can be downloaded from http://www.ipcc-data.org/sim/gcm_global/index.html.12For a precise description of the storylines and scenarios see Carter (2007), http://www.ipcc-data.
org/guidelines/TGICA_guidance_sdciaa_v2_final.pdf.
90 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
temperature growth (medium temperature increase); the commit group reflects
scenarios of very low temperature growth (low temperature increase). The
measurements of the original series are in degrees Celsius [°C].
3.4 Forecasting
Model comparison
To assess the forecasting power of the state-space models proposed in Section 3.2,
we carry out the following forecasting exercise. Let n denote the complete sample
size (i.e. n = 1560, corresponding to monthly observations ranging from January
1880 to December 2009), n∗ < n the size of the estimation sample, h the forecast
horizon, f = n −n∗ the number of forecasts for a given estimation sample size. For
all forecasting methods, the setup of the exercise is the following:
1. estimate the system parameters using observations of sea level and tempera-
ture from time t = 1 up to t = n∗;
2. compute forecasts using temperature observations from time t = n∗ + 1 to
t = n.
In particular, for the state-space system (3.34) (and for the model with quadratic
trend) the steps to construct the forecasts are the following :
(i) estimate the system parameters by maximum likelihood, using observations of
sea level and temperature from time t = 1 up to t = n∗ (the likelihood function
is delivered by the Kalman filter, see for instance Durbin and Koopman (2012));
(ii) run the Kalman filter, using the estimated parameters, on the dataset composed
of observations of the sea level from time t = 1 to t = n∗ and observations of
the temperature from t = 1 to t = n.
(iii) the forecasts of the sea level are then the filtered values Sn∗+h with h = 1, . . . , f .
Observations of the sea level, from time t = n∗+1 to t = n, are treated as missing
values, see Durbin and Koopman (2012) for more details on how to modify the filter
in case of missing observations. Note that for a linear state-space system the Kalman
filter delivers the best linear predictions of the state vector, conditionally on the
observations. Moreover, if the innovations are Gaussian, the filtered states coincide
with conditional expectations, for more details on the optimality properties of the
Kalman filter see Brockwell and Davis (2009).
We select two benchmark forecasting methods to which we compare our specifica-
tions. In particular, we compare our model to the procedures developed in Rahmstorf
(2007b) and Vermeer and Rahmstorf (2009). The choice of these benchmarks reflects
their popularity in the literature and the replicability of the results in the papers due
to the availability of source codes.
3.4. FORECASTING 91
Rahmstorf (2007b) procedure
The first competing method is the one used in Rahmstorf (2007b). The model is based
on equations (3.14)-(3.15). The procedure can be summarized as follows:
(i) the sea-level and temperature series, from time t = 1 up to t = n∗, are smoothed
using singular spectrum analysis and embedding dimension equal to ned = 180
months, corresponding to 15 years (15×12 = 180), as used in their paper;
(ii) first differences of the smoothed sea-level series are then taken;
(iii) the smoothed series (temperature and first differences of the sea level) are then
divided into nbi n = 60 months bins, corresponding to 5 years (5×12 = 60), as
used in the paper, and in each bin the average is taken;
(iv) the time series of bin-averages are then detrended (fitting a linear trend);
(v) the bin-averages of sea level in first differences is then regressed onto the
bin-averages of the temperature in levels;
(vi) the estimated regression coefficients are then used to compute the values of the
sea level in first differences from the out-of-sample (smoothed) temperatures
(note that the information set used comprises the sea-level observations from
time t = 1 to t = n∗, and observations of the temperature from t = 1 to t = n);
(vii) the forecasts of the sea level are then obtained by summing the forecast sea
level in first differences.
We also compute forecasts with the combinations: ned = 60/nbi n = 60, ned = 60/nbi n =180, and ned = 180/nbi n = 180.
Vermeer and Rahmstorf (2009) procedure
The second competing method is the one used in Vermeer and Rahmstorf (2009).
Their model is based on equation (3.16). The procedure is similar to the previous one,
with the addition of an extra step, and it can be summarized as follows:
(i) the sea-level and temperature series, from time t = 1 up to t = n∗, are smoothed
using singular spectrum analysis and an embedding dimension equal to ed =180 months, corresponding to 15 years (15×12 = 180), as used in the paper;
(ii) first differences of the smoothed sea-level series are then taken;
(iii) the smoothed series (temperature and first differences of the sea level) are then
divided into 60 months bins, corresponding to 5 years (5×12 = 60), as used in
their paper, and in each bin the average is taken;
(iv) both time series of bin-averages are then detrended (by fitting a linear trend);
92 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
(v) the parameter v2, in equation (3.16) (λ in the notation of their paper), is then
selected as the value for which the correlation between the detrended bin-
averages of the smoothed temperature and the detrended bin-averages of the
first differences of the smoothed sea level, is maximized;
(vi) the bin-averages of the smoothed sea level in first differences are then regressed
on the bin-averages of the smoothed temperature in levels, corrected for the
rate of change of the temperature (that is the v2dT (t )
d t factor in equation (3.16));
(vii) the estimated regression coefficients are then used to compute the values of the
sea level in first differences from the out-of-sample (smoothed) temperatures,
corrected for the v2dT (t )
d t factor in equation (3.16), where the v2 used is the one
previously computed and the rate of change dT (t )d t is computed in the same way
as before but from the out-of-sample (smoothed) temperatures (note that the
information set used comprises the sea-level observations from time t = 1 to
t = n∗ and observations of the temperature from t = 1 to t = n);
(viii) the forecasts of the sea level are then obtained by summing the forecast sea
level in first differences.
We also compute forecasts with the combinations: ned = 60/nbi n = 60, ned = 60/nbi n =180, and ned = 180/nbi n = 180.
Performance measure
As a measure of the relative forecasting power between the models, we take the ratios
of the square roots of the mean squared forecast errors, from the different models.
For model j we have:
R jf =
√√√√ 1
f
f∑h=1
(S j
n∗+h −Srn∗+h
)2, (3.35)
where S jn∗+h denotes the sea-level forecast from the j -th model and Sr
n∗+h is the
observed sea level13. We select different values of n∗. Note that in computing R jf we
are giving equal weight to forecasts at different horizons. This strategy is motivated by
the final goal of the model, that is to make long-term projections of the sea level. To
compare the models, we take ratios between the R measures, equation (3.35), from
the different forecasting models/methods.
13The superscript r stands for “reconstructed”, as the observations are a reconstruction of the sea-levelseries, made from different measurements of the sea level.
3.4. FORECASTING 93
Forecasting conditional on AR4-IPCC temperature scenarios
In this subsection we explain the method used to make long-term projections for
the sea level, conditional on the IPCC temperature scenarios. First, note that the
measurements of sea level and temperature go from January 1880 to December
2009 and correspond to monthly averages, whereas the IPCC temperature scenarios,
ranging from 2010 to 2099, correspond to yearly values. In order to model the data
and the scenarios in the same framework, we transform the yearly values in monthly
ones. In particular, for each scenario we treat the temperature value corresponding
to a specific year, as an observation for the month of July for that year, treating the
values for the remaining months as missing values.
Denote with ntot the sample size of the assembled dataset made up of the monthly
observations of sea level and temperatures plus the IPCC temperature scenarios, in
particular we have ntot = 1560+1080 = 2640. To construct the sea-level forecasts,
conditional on the temperature scenarios, we follow these steps:
(i) estimate the system parameters by maximum likelihood using observations of
sea level and temperature from time t = 1 up to time t = n;
(ii) run the Kalman filter, using the estimated parameters, on the dataset composed
of observations of the sea level and temperature from time t = 1 to t = n, and
one temperature scenario from t = n +1 to t = ntot ;
(iii) the forecasts of the sea level are then the smoothed values Sn+h with h =1, . . . ,ntot −n.
The procedure is repeated for each of the 75 temperature scenarios. In order to
compute confidence intervals for the sea-level projections, we first use a bootstrap
procedure to obtain an empirical distribution function (EDF) for the forecasts, condi-
tioning on each scenario separately. We then aggregate these EDFs using the law of
total probabilities, assigning equal probability to the different scenarios. Denoting
with Bi the i -th IPCC temperature scenario, where i = 1, . . . , N (N = 75) and with h the
forecast horizon, the unconditional empirical distribution function for the sea-level
projections is
Pr(St+h ≤ s
)=
N∑i=1
Pr(St+h ≤ s|Bi
)Pr
(Bi
)= 1
N
N∑i=1
Pr(St+h ≤ s|Bi
), (3.36)
where Pr (St+h ≤ s|Bi ) is the conditional distribution function of the sea-level fore-
cast St+h , given a temperature scenario Bi , and Pr (Bi ) is the probability of the i -th
temperature scenario. The confidence intervals for the forecasts St+h are then ob-
tained by taking the 1st and 99th percentiles of the cumulative distribution function
94 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Pr (St+h ≤ s). The EDFs Pr (St+h ≤ s|Bi ) are obtained using a bootstrap procedure,
and the probabilities Pr (Bi ) are set equal to 1/N . We thus assume equal probabilities
for the different temperature scenarios, in line with the literature.
The bootstrap procedure is detailed in Appendix 3.9 and it is a modification of the
method proposed in Rodriguez and Ruiz (2009). In Rodriguez and Ruiz (2009), they
consider a time invariant state-space system in which the system parameters do not
vary in time. In the present paper, however, we assume a time-varying measurement
noise variance for the sea level σ2,St (3.33), this introduces heteroskedasticity in the
innovations.
3.5 Computational aspects
The parameters of the state-space system are estimated by maximum likelihood.
The likelihood function is delivered by the Kalman filter. We employ the univariate
Kalman filter derived in Koopman and Durbin (2000), as we assume a diagonal
covariance matrix for the innovations in the measurement equation. The maximum
of the likelihood function has no explicit form solution and numerical methods have
to be employed. We make use of two algorithms:
• CMA-ES. Covariance Matrix Adaptation Evolution Strategy, see Hansen and
Ostermeier (1996)14. This is a genetic algorithm that samples the parameter
space according to a Gaussian search distribution, which changes according to
where the best solutions are found in the parameter space;
• BFGS. Broyden-Fletcher-Goldfarb-Shanno, see for instance Press et al. (2002).
This algorithm belongs to the class of quasi-Newton methods and requires the
computation of the gradient of the function to be minimized.
The CMA-ES algorithm performs very well when no good initial values are available
but it is slower to converge than the BFGS routine. The BFGS algorithm, on the
other hand, requires good initial values but converges considerably faster than the
CMA-ES algorithm (once good initial values have been obtained). Hence, we use the
CMA-ES algorithm to find good initial values and then the BFGS one to perform the
minimizations with the different sample sizes, needed in the forecasting exercise
detailed in Section 3.4.
To gain speed we choose C++ as the programming language, using routines from
the Numerical Recipes, Press et al. (2002). We compile and run the executables on a
Linux 64-bit operating system using the GCC compiler 15. The integrals appearing in
equations (3.31) can be computed analytically with the aid, for instance, of MATLABr
14See https://www.lri.fr/~hansen/cmaesintro.html for references and source codes. The au-thors provide C source code for the algorithm which can be easily converted into C++ code.
15See http://gcc.gnu.org/onlinedocs/ for more information on the Gnu Compiler Collection,GCC.
3.6. RESULTS AND DISCUSSION 95
symbolic toolbox. The generated code can then be directly converted into C++ code
with the command ccode.
3.6 Results and discussion
Model comparison results
To compute the out-of-sample forecasts for model (3.34) we use the Kalman filter,
treating the sea-level values as missing observations, for the time points at which we
want to forecast it. In the tables 3.1-3.4 in Appendix 3.9 are reported the ratios (3.35)
for different values of the estimation sample n∗ and forecast sample f . We label the
different forecasting models according to the following convention.
• model 1. State-space system (3.22) with linear trend (3.24), taking the filtered
values as forecasts;
• model 2. State-space system (3.22) with linear trend (3.24), taking the smoothed
values as forecasts;
• model 3. State-space system (3.22) with quadratic trend (3.25), taking the fil-
tered values as forecasts;
• model 4. State-space system (3.22) with quadratic trend (3.25), taking the
smoothed values as forecasts;
• model 5. Rahmstorf (2007b) procedure (see Section 3.4) with embedding di-
mension ned = 60 and number of bins nbi n = 60;
• model 6. Rahmstorf (2007b) procedure (see Section 3.4) with embedding di-
mension ned = 60 and number of bins nbi n = 180;
• model 7. Rahmstorf (2007b) procedure (see Section 3.4) with embedding di-
mension ned = 180 and number of bins nbi n = 60;
• model 8. Rahmstorf (2007b) procedure (see Section 3.4) with embedding di-
mension ned = 180 and number of bins nbi n = 180;
• model 9. Vermeer and Rahmstorf (2009) procedure (see Section 3.4) with em-
bedding dimension ned = 60 and number of bins nbi n = 60;
• model 10. Vermeer and Rahmstorf (2009) procedure (see Section 3.4) with
embedding dimension ned = 60 and number of bins nbi n = 180;
• model 11. Vermeer and Rahmstorf (2009) procedure (see Section 3.4) with
embedding dimension ned = 180 and number of bins nbi n = 60;
96 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
• model 12. Vermeer and Rahmstorf (2009) procedure (see Section 3.4) with
embedding dimension ned = 180 and number of bins nbi n = 180.
It can be seen from tables 3.1-3.4 that the state-space models 1-2 and 3-4 perform
quite well compared to models 5-12. In particular, the quadratic trend component (in
models 3-4) seems to help the forecasting performance of the state-space system. The
difference between the filtered and smoothed forecasts is negligible. These results
show that similar (or better) forecasts can be obtained using the state-space systems
presented in Section 3.2, compared to the two benchmark procedures outlined in
Section 3.4.
We also considered specifications without trend components, with stochastic
trends, and/or with various sets of coefficients restricted to zero. All these alternative
specifications were found to perform poorly compared to the ones presented in
this paper, in terms of forecasting performance. In particular, setting the coefficient
aSS = 0 or adding stochastic components in the trend process, considerably worsened
the forecast performance of the models.
Full sample estimation results
In this subsection we present the parameter estimates relative to models 1-2 and
models 3-4. The estimation results are contained in tables 3.5-3.6 (for models 1-
2) and in tables 3.7-3.9 (for models 3-4) and they are divided into estimates for
the continuous-time and discrete-time specifications. The tables can be found in
Appendix 3.9. We estimate the parameters using the complete dataset of sea-level and
temperature monthly observations, ranging from January 1880 to December 2009,
for a sample size equal to 1560. We group the comments according to the different
models (1-2 and 3-4):
• models 1-2. The estimated standard deviation of the temperature measure-
ment error is σT = 7.59[cK ] (0.0759[K ]), which is slightly lower than the aver-
age standard deviation of the sea-level measurement errors σS = 11.43[mm]
(0.01143[m]). The average σS is computed from the sequence of volatilities
σSττ=1:n , corresponding to the uncertainty estimates reported in Church and
White (2011). The autoregressive coefficients A∗,SS = 0.99 and A∗,T T = 0.92 are
both close to unity. The coefficient linking sea level to temperature is found
to be quite small, A∗,ST = 0.0054[mm/cK ] (0.00054[m/K ]), whereas the one
linking temperature to the sea level A∗,T S = 0.0489[cK /mm] (0.489[K /m]) is
quite large.
• models 3-4. The estimated standard deviation of the temperature measure-
ment error isσT = 7.41[cK ] (0.0741[K ]). The autoregressive coefficients A∗,SS =0.97 and A∗,T T = 0.89 are both close to unity. Interestingly, the coefficients
linking sea level to temperature, and vice versa, are found to have a negative
3.6. RESULTS AND DISCUSSION 97
sign A∗,ST = −0.0035[mm/cK ] (−0.00035[m/K ]), A∗,T S = −0.0426[cK /mm]
(−0.426[K /m]).
The parameter aST , if left unrestricted, is estimated to be either positive or neg-
ative (depending on the model and the estimation sample) and of the order of
10−2[mm/cK ] (10−3[m/K ]) for a one month time-step. The low value of this pa-
rameter may be caused by long response times of the sea level to the temperature
and the fact that the time-step considered is quite small. Early studies indicate lags
in the order of 20 years, between temperature and sea-level rise, Gornitz et al. (1982).
One puzzling fact is the change in sign of aST and aT S between models 1-2 and 3-4.
We found that when the parameter aSS is left unconstrained, the coefficient link-
ing sea level and temperature is estimated to be very low. This may suggest long
response times of the sea level to the temperature changes, possible distortions in
the sea level and temperature reconstructions, and/or a misspecification of the func-
tional link between the two variables.
One interesting finding concerns the role of the sea-level measurement error vari-
anceσ2,Sτ in the state-space system. Namely, this parameter regulates the smoothness
of the filtered (and smoothed) sea-level series. Interestingly, if σ2,Sτ =σ2,S is left unre-
stricted and estimated together with the other parameters, the value obtained is very
close to zero. This causes the filtered (and smoothed) sea-level series to essentially
coincide with the observed ones. Intuitively, in this case the forecasts worsen.
Forecasting conditional on AR4-IPCC scenarios results
In this subsection we report the long-term sea-level projections, computed condition-
ally on the different temperature scenarios. See Section 3.4 for more details on the
forecasting procedure and Section 3.3 for a description of the temperature scenarios
used. The scenarios are depicted in figure 3.1. We make sea-level rise projections
using models 2 and 4, with respect to the (smoothed) sea-level value in 2009:
• model 2. The forecasts are quite sensitive to the temperature scenarios. De-
note with q0.01 and q0.99 the 1st and 99th percentiles of the distribution of
sea-level forecasts for the year 2099. If we condition on all of the 75 temper-
ature scenarios, taken with equal probability, the sea-level forecasts range
from q0.01 = 0.0948[m] to q0.99 = 0.3525[m] with a mean value of 0.2130[m],
see figure 3.2. This range changes if different scenario groups are considered
separately.
(i) A1b group: forecasts range from q0.01 = 0.1829[m] to q0.99 = 0.3537[m]
with a mean value of 0.2470[m];
(ii) A2 group: forecasts range from q0.01 = 0.2030[m] to q0.99 = 0.3654[m]
with a mean value of 0.2845[m];
98 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
(iii) B1 group: forecasts range from q0.01 = 0.1485[m] to q0.99 = 0.2688[m]
with a mean value of 0.1941[m];
(iv) commit group: forecasts range from q0.01 = 0.0868[m] to q0.99 = 0.1489[m]
with a mean value of 0.1174[m].
• model 4. The forecasts are not very sensitive to the temperature scenarios. In
particular, the forecasts relative to 2099 for the sea level range between q0.01 =0.1999[m] and q0.99 = 0.2817[m], with a mean value of 0.2410[m], conditioning
on all temperature scenarios, taken with equal probability, see figure 3.3. This
range does not change much if different groups are considered separately.
(i) A1b group: forecasts range from q0.01 = 0.1991[m] to q0.99 = 0.2696[m]
with a mean value of 0.2375[m];
(ii) A2 group: from q0.01 = 0.1886[m] to q0.99 = 0.2704[m] with a mean value
of 0.2341[m];
(iii) B1 group: from q0.01 = 0.2121[m] to q0.99 = 0.2738[m] with a mean value
of 0.2428[m];
(iv) commit group: from q0.01 = 0.2072[m] to q0.99 = 0.2944[m] with a mean
value of 0.2507[m].
The difference between the average, smoothed sea level in 1990 and the smoothed
sea level in December 2009 is 0.0549[m]. Consequently, to compute the sea-level
changes with respect to the average 1990 level, 0.0549[m] has to be added to the
previous values. We make this remark because in several papers the sea-level rise
forecasts are reported with respect to the 1990 average, e.g. in Rahmstorf (2007b),
Vermeer and Rahmstorf (2009), and Grinsted et al. (2010).
3.6. RESULTS AND DISCUSSION 99
Figure 3.1. AR4-IPCC tempterature scenarios.
1900 1950 2000 2050 2100
−1
0
1
2
3
4
years
temperature
anomalies(K
)
global mean temperature observationsAR4-IPCC-a1b scenariosAR4-IPCC-a2 scenariosAR4-IPCC-b1 scenariosAR4-IPCC-commit scenarios
Temperature observations and AR4-IPCC-SRES tempterature scenarios. The temperature observations areyearly averages, ranging from January 1880 to December 2009. The AR4-IPCC-SRES scenarios correspondto yearly simulated values, ranging from January 2010 to December 2099.
Figure 3.2. Forecasts based on A1b-A2-B1-commit-IPCC scenarios and model 2.
1900 1950 2000 2050 2100
−0.2
−0.1
0
0.1
0.2
0.3
years
sea-level
change(m
)
sea-level observationssea-level smoothed series and projections0.98 confidence interval
Forecasts based on IPCC-SRES (A1b, A2, B1, and commit groups) scenarios and model 2 (see Section3.4). The observations and the projections are monthly values. The sea-level observations range fromJanuary 1880 to December 2009. The projections correspond to smoothed monthly values and range fromJanuary 2010 to December 2099. The base sea-level value is the smoothed sea level in December 2009. Theconfidence bands correspond to a 98% confidence interval (see Section 3.4).
100 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Figure 3.3. Forecasts based on A1b-A2-B1-commit-IPCC scenarios and model 4.
1900 1950 2000 2050 2100
−0.2
−0.1
0
0.1
0.2
0.3
years
sea-level
change(m
)
sea-level observationssea-level smoothed series and projections0.98 confidence interval
Forecasts based on IPCC-SRES (A1b, A2, B1, and commit groups) scenarios and model 4 (see Section3.4). The observations and the projections are monthly values. The sea-level observations range fromJanuary 1880 to December 2009. The projections correspond to smoothed monthly values and range fromJanuary 2010 to December 2099. The base sea-level value is the smoothed sea-level in December 2009. Theconfidence bands correspond to a 98% confidence interval (see Section 3.4).
3.7 Conclusions
In this paper we proposed a statistical framework to model and forecast the global
mean sea level, conditional on the global mean temperature. The specification is for-
mulated as a continuous-time state-space system. The state vector is composed of the
unobserved sea level and temperature processes, as well as trend components and,
jointly, follow an Ornstein-Uhlenbeck process. This process can be exactly discretised.
The measurement equation adds independent noise to the discretely sampled states.
The resulting system is linear and Kalman filtering techniques apply. In particular,
the Kalman filter is used to compute the likelihood function. Furthermore, we exploit
the ability of the Kalman filter to deal with missing observations, to make projections
for the sea level. The state-space specification also allows to model changes in the
accuracy of the reconstructed sea-level series. Specifically, this is achieved by allowing
the volatility parameter of the sea-level measurement error to be time-varying, and
matching it to the sea-level uncertainty values reported in Church and White (2011).
We find that this modelling scheme performs better, in forecasting, compared to one
in which the volatility of the measurement error of the sea level is estimated together
with the other parameters. If σ2,Sτ =σ2,S is left unrestricted and estimated together
with the other parameters, the value obtained is very close to zero. This causes the
filtered (and smoothed) sea-level series to essentially coincide with the observed one.
In this case the predictive ability of the model deteriorates. The advantage of using
the proposed state-space model is that there is no difference between the system
dynamics assumed for the variables of interest and the statistical model estimated
3.7. CONCLUSIONS 101
using real data.
The choice of the models was made according to their forecasting performance,
relative to selected benchmarks, namely the Rahmstorf (2007b) and Vermeer and
Rahmstorf (2009) methods. This model selection criterion was chosen because of the
final objective of semi-empirical models, that is making long-term projections for
the sea level.
We find that the magnitude of the parameter A∗,ST , linking the sea level to the
temperature, is estimated to be of the order of 10−2[mm/cK ] (10−3[m/K ]). Note
that A∗,ST relates the value of the unobserved temperature process at time τ to the
value of the sea level at time τ+1, where the time step corresponds to one month.
The low value of A∗,ST may be caused by long response times of the sea level to the
temperature and the fact that the time step is quite short. Early studies indicate lags
of the order of 20 years (240 months), between temperature and sea-level rise, see for
instance Gornitz et al. (1982).
When the parameter aSS is left unconstrained, the coefficient linking sea level and
temperature is estimated to be very low. However, if the parameter aSS is restricted to
be zero, the forecast performance of the model deteriorates considerably. One puz-
zling fact is the change in sign of A∗,ST and A∗,T S , between the linear and quadratic
trend specifications. In particular, both parameters are positive in the linear trend
specification and negative in the quadratic trend one. This finding is quite surprising,
considering that the quadratic trend model seems to forecast better than the linear
one. Concerning the model comparison exercise, the state-space specifications be-
have well compared to the Rahmstorf (2007b) and Vermeer and Rahmstorf (2009)
methods. The choice of the trend component influences somewhat the forecasting
performance of the model.
We make projections for the sea level from 2010 up to 2099. Under the linear trend
specification the forecasts are quite sensitive to the temperature scenarios, whereas
under the quadratic one the projections are quite similar across scenarios. Denote
with q0.01 and q0.99 the 1st and 99th percentiles, respectively, of the distribution of
sea-level rise forecasts for the year 2099 with respect to the (smoothed) sea-level value
in 2009. Conditionally on all the 75 temperature scenarios, the sea-level rise forecasts
range from q0.01 = 0.0948[m] to q0.99 = 0.3525[m], with a mean value of 0.2130[m]
under the linear trend specification, and from q0.01 = 0.1999[m] to q0.99 = 0.2817[m],
with a mean value of 0.2410[m], under the quadratic trend model. With respect to
the mean, smoothed 1990 sea-level value, the above results translate into sea-level
rise forecasts ranging from q0.01 = 0.1497[m] to q0.99 = 0.4074[m], with a mean value
of 0.2679[m] under the linear trend specification, and from q0.01 = 0.2548[m] to
q0.99 = 0.3366[m], with a mean value of 0.2959[m], under the quadratic trend model.
The projections obtained with the models proposed in this study are lower than
the ones obtained in Rahmstorf (2007b), Vermeer and Rahmstorf (2009), and Grin-
sted et al. (2010). In particular, their projections for the year 2100, with respect to the
102 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
mean sea level in 1990, range from 0.5[m] to 1.38[m] for Rahmstorf (2007b), from
0.72[m] to 1.81[m] for Vermeer and Rahmstorf (2009), and from 1.30[m] to 1.80[m]
(or between 0.95[m] to 1.48[m], depending on the temperature reconstruction) for
Grinsted et al. (2010). Note, however, that in these three papers the interpretation of
the range spanned by the forecasts is different from ours. Most likely, we obtain lower
estimates for the sea level because of the coefficient relating temperature to sea-level
A∗,ST , which in the state-space specifications is estimated to be quite small. This im-
plies longer estimated response times of the sea level to changes in the temperature,
with respect to the aforementioned studies.
Possible continuations of this work could be represented by considering alter-
native continuous-time stochastic processes to model sea level and temperature,
for instance geometric Brownian motions or more general Itô processes. Another
important aspect in this analysis was the specification of the trend components,
which played a key role in the forecasting of the sea level. It would be interesting
to study alternative specifications for the trend components and their relation to
different climate forcings, such as human induced changes in greenhouse gases and
aerosols concentrations.
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3.9. APPENDIX 107
3.9 Appendix
Details of univariate Kalman filter
In this section we give details on the Kalman filter and its univariate version. The
univariate Kalman filter was used for the computation of the likelihood function and
for the bootstrap procedure, needed to compute prediction intervals for the sea-level
projections, see Section 3.6. This subsection draws heavily on Durbin and Koopman
(2012).
Consider the following state-space system
yt = Zαt +εt ,
αt+1 = c+Tαt +ηt , (3.37)
where εt ∼ N (0,Ht ) takes values in Rp , with Ht a covariance matrix, ηt ∼ N (0,Q)
takes values in Rk , with Q a covariance matrix, yt ∈ Rp , αt ∈ Rk , and Z, T, and c are
parameter matrices and vectors of appropriate dimensions. Notice that the state-
space system (3.34) is of the same type as system (3.37). The standard Kalman filter
recursions for system (3.37) are
vt = yt −Zat ,
Ft = ZPt Z′+Ht ,
Kt = Pt Z′, (3.38)
at |t = at + Kt F−1t vt , Pt |t = Pt − Kt F−1
t K′t ,
at+1 = Tt at |t +c, Pt+1 = TPt |t T′+Q, (3.39)
for t = 1, . . . ,n, where Pt = E[[αt −at ][αt −at ]′
], Pt |t = E
[[αt −at |t ][αt −at |t ]′
], and
at = E [α|y0, . . . ,yt−1] and at |t = E [α|y0, . . . ,yt ] are the one-step-ahead prediction and
the filtered states, respectively. Koopman and Durbin (2000) derived a univariate
version of this algorithm in the case of diagonal variance-covariance matrices Ht . In
this case the system (3.37) can be represented as
yt ,i = ziαt ,i +εt ,i , t = 1, . . . ,n i = 1, . . . , p,
αt ,i+1 =αt ,i , t = 1, . . . ,n i = 1, . . . , p −1,
αt ,i+1 = c+Tαt ,i +ηt , t = 1, . . . ,n i = p. (3.40)
Where zi is the i − th row of matrix Z, yt ,i and εt ,i ∼ N (0,σ2t ,i ) are the i − th compo-
nents of yt and εt , respectively. The Kalman filter recursions for specification (3.40)
108 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
can be written as
vt ,i = yt ,i −zi at ,i , (3.41)
Ft ,i = zi Pt ,i z′i +σ2t ,i , (3.42)
kt ,i = Pt ,i z′i , (3.43)
at ,i+1 = at ,i + kt ,i F−1t ,i vt ,i
Pt ,i+1 = Pt ,i − kt ,i F−1t ,i k
′t ,i
for i = 1, . . . , p −1 t = 1, . . . ,n, (3.44)
at+1,1 = T(at ,i + kt ,i F−1t ,i vt ,i )+c
Pt+1,1 = T(Pt ,i − kt ,i F−1t ,i k
′t ,i )T′+Q
for i = p t = 1, . . . ,n. (3.45)
Notice that Ft ,i is a scalar. As a consequence the univariate recursions do not require
the inversion of p ×p matrices, as in the standard Kalman filter recursions (3.38)-
(3.39) and can lead to computational savings.
The state-space system (3.40) has two types of disturbances, namely εt ,i , and ηt .
The so called “innovation form” has a unique source of disturbance, that is vt ,i . The
innovation form is made up of the following equations:
yt ,i = zi at ,i + vt ,i ,
kt ,i = Pt ,i z′i ,
at ,i+1 = at ,i +kt ,i F−1t ,i vt ,i , for i = 1,2, . . . , p −1 t = 1,2, . . . ,n,
at+1,1 = T(at ,i +kt ,i F−1
t ,i vt ,i
)+c, for i = p t = 1,2, . . . ,n. (3.46)
The innovation form (3.46) of the state-space system (3.40) constitutes the basis for
the bootstrap procedure outlined in the following subsection.
3.9. APPENDIX 109
Bootstrap procedure
In this section we outline the bootstrap procedure used to compute the prediction
intervals for the sea-level projections, conditional on the IPCC scenarios, as described
in Section 3.6. We detail the algorithm with respect to the state-space system (3.37)
and the Kalman filter recursions (3.41)-(3.45). The algorithm is a modification of the
one proposed in Rodriguez and Ruiz (2009) that allows for time-varying measurement
error variances. Denote withθ the vector containing the parameters of the state-space
system (3.37). The algorithm we propose is made up of the following steps:
1. estimate the parameters of model (3.30) by maximum likelihood and obtain θ
and the sequence of innovations vt ,i i=1,...,pt=1,...,n ;
2. compute the centred innovations vct ,i i=1,...,p
t=1,...,n , obtained as vct ,i = vt ,i − vn,i , with
vn,i = (1/n)∑n
t=1 vt ,i ;
3. obtain the standardized innovations v st ,i i=1,...,p
t=1,...,n , computed as v st ,i =
vct ,ipFt ,i
;
4. obtain a sequence of bootstrap standardized innovations v∗t ,i i=1,...,p
t=1,...,n via ran-
dom draws with replacement from the randomly scaled standardized innova-
tions v st ,i ·εt ,i i=1,...,p
t=1,...,n , where εt ,i ∼ N (0,1);
5. compute a bootstrap replicate of the observations y∗t ,i i=1,...,p
t=1,...,n by means of the
innovation form (3.46) using v∗t ,i i=1,...,p
t=1,...,n and the estimated parameters θ;
6. estimate the corresponding bootstrap parameters θ∗ from the bootstrap repli-
cates;
7. run the Kalman filter with θ∗ using the original observations and one tempera-
ture scenario as described in Section 3.4.
Steps 1-7 are repeated N = 500 for each temperature scenario. As made clear in step
4 we make use of a wild bootstrap procedure as opposed to the simple re-sampling
method used in Rodriguez and Ruiz (2009). The wild bootstrap was originally pro-
posed by Wu (1986) and it is well known in the literature to perform better than a
simple resampling scheme in the presence of heteroskedasticity, see for instance
Liu et al. (1988) and Mammen (1993). In this paper the heteroskedasticity comes
from the time varying matrix Ht . Note that the variance of the innovations vt ,i is
given by Ft ,i = zi Pt ,i z′i +σ2t ,i where σ2
t ,i is time-varying. In the notation of equation
(3.33), it’s the parameter σ2,St that causes the innovations vt ,i , in equation (3.41), to
be heteroskedastic.
110 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
State-space system and dimensional analysis
In this section we rewrite the state-space system (3.22) with linear trend (3.24), making
clear the fundamental dimensions and the units of measurement of the quantities
involved. We make use of SI units apart from the time dimension, for which we use
months (or years). The time series of the sea level is in millimetres [mm] and the
temperature one is in centikelvin [cK ].
Continuous-time state equation
The continuous-time process driving the state equation has the following dimensions:
d
[mm]S(t )[cK ]T (t )[mm]
µS (t )[cK ]
µT (t )
=
0
0[mm
month
]λS[mm
month
]λT
[month]
d t
+
[1
month
]aSS
[mm
cK ·month
]aST
[1
month
]κS 0[
cKmm·month
]aT S
[1
month
]aT T 0
[1
month
]κT
0 0 0 0
0 0 0 0
[mm]S(t )[cK ]T (t )[mm]
µS (t )[cK ]
µT (t )
[month]
d t
+
1 0
0 1
0 0
0 0
[mm]
dηS (t )[cK ]
dηT (t )
, (3.47)
denoting with dη(t ) = [dηS (t ) : dηT (t )], we have E [dη(t )dη(t )′] =Σd t , withΣ a sym-
metric positive semidefinite matrix. To understand the units of measurement of the
components of Σ we can reason in the following way: first, write E [dη(t)dη(t)′] =E [
pΣdW(t)dW(t)′
pΣ′] where
pΣ represents a square root of the matrix Σ and
dW(t ) = [dW S (t ) : dW T (t )]′ is a two-dimensional Brownian motion such that
E [dW(t )dW(t )′] =[
1 ρ
ρ 1
]d t , (3.48)
where |ρ| < 1; second, denote
pΣ =
[ωSS ωST
ωT S ωT T
], (3.49)
3.9. APPENDIX 111
we can write
pΣdW(t ) =
[ωSS dW S (t )+ωST dW T (t )
ωT S dW S (t )+ωT T dW T (t )
], (3.50)
and
E[pΣdW(t )dW(t )′
pΣ′] =
[ΣSS ΣST
ΣT S ΣT T
]d t , (3.51)
where
ΣSS d t = E[
(ωSS dW S )2 + (ωST dW T )2 +2ωSSωST dW S dW T]
= (ωSS )2d t + (ωST )2d t +2ωSSωSTρd t
=[
(ωSS )2 + (ωST )2 +2ωSSωSTρ]
d t , (3.52)
and
ΣST d t = E[ωSSωT S (dW S (t ))2 +ωSTωT S dW S (t )dW T (t )
]+ E
[ωSSωT T dW S (t )dW T (t )+ωSTωT T dW T (t )dW T (t )]
]= ωSSωT S d t +ωSTωT Sρd t
+ ωSSωT Tρd t +ωSTωT T d t
=[ωSSωT S +ωSTωT Sρ+ωSSωT Tρ+ωSTωT T
]d t , (3.53)
from equation (3.52) we can deduce that ωSS and ωST have units of measurement
corresponding to [mm/p
month]; to see this set ωST = 0 in equation 3.52, we have
then ΣSS d t = (ωSS )2d t which has units of measurement of [mm2], as it is the expec-
tation of the square of a quantity with units [mm], this implies that (ωSS )2 and ΣSS
have units of [mm2/month], as time is measured in months [month]; from equation
(3.52) it is also clear that ωST has the same units as ωSS . Using the same line of rea-
soning we deduce that (ωT T )2, (ωT S )2, and ΣT T have units of [cK 2/month]. Having
recovered the units of measurement of ωSS , ωST , ωT S , and ωT T we can deduce from
equation (3.53) that the units of measurement ofΣST andΣT S are [(mm ·cK )/month].
In summary we obtain:
Σ =
[mm2
month
]ΣSS
[mm·cKmonth
]ΣST[
cK ·mmmonth
]ΣT S
[cK 2
month
]ΣT T
. (3.54)
112 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Discrete-time state vector and measurement equation
The discretized state vector has the following dimensions:
[mm]Sτ+1[cK ]
Tτ+1[mm]
µSτ+1
[cK ]
µTτ+1
=
[mm]
c∗,S
[cK ]
c∗,T
[mm]
c∗,µS
[cK ]
c∗,µT
+
A∗,SS
[mmcK
]A∗,ST A∗,SµS
[mmcK
]A∗,SµT
[cK
mm
]A∗,T S A∗,T T
[cK
mm
]A∗,TµS
A∗,TµT
0 0 1 0
0 0 0 1
[mm]Sτ
[cK ]Tτ
[mm]
µSτ
[cK ]
µTτ
+
[mm]
ξSτ
[cK ]
ξTτ
0
0
, (3.55)
denoting ξτ = [ξSτ : ξT
τ ]′, we have
E[ξτξ
′τ
]=
[
mm2]
Σ∗,SS[mm·cK ]Σ∗,ST
[cK ·mm]Σ∗,T S
[cK 2
]Σ∗,T T
, (3.56)
where the units of measurement were obtained by following the same logic as in the
previous section.
Finally, the measurement equation has the following dimensions:
[mm]Srτ
[cK ]T rτ
=[
1 0 0 0
0 1 0 0
]
[mm]Sτ
[cK ]Tτ
[mm]
µSτ
[cK ]
µTτ
+
[mm]
εSτ
[cK ]
εTτ
. (3.57)
3.9. APPENDIX 113
Tables
In this section we report the tables with the results for the model comparison fore-
casting exercise and the parameter estimates for models 1-2 and models 3-4. See
Sections 3.4 and 3.6 for more details.
114 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Table
3.1.Ratio
sb
etween
R1f
and
Rjf ,
j=1,...,12.
n ∗1535
15101485
14601435
14101385
13601335
1310f
2550
75100
125150
175200
225250
mo
del1
1,001,00
1,001,00
1,001,00
1,001,00
1,001,00
mo
del2
1,001,02
1,021,00
1,011,04
0,891,06
1,031,02
mo
del3
0,991,10
1,081,03
1,011,03
1,021,00
1,021,03
mo
del4
0,991,09
1,071,02
0,990,99
1,020,9
0,971,03
mo
del5
1,071,00
1,311,21
1,191,24
1,060,75
1,091,04
mo
del6
1,060,98
1,261,17
1,070,82
0,621,01
0,560,5
mo
del7
0,71,26
1,241,09
1,171,14
0,760,55
1,010,97
mo
del8
0,71,25
1,221,08
1,151,22
1,191,15
1,030,9
mo
del9
1,071,00
1,311,21
1,191,24
1,060,75
1,091,04
mo
del10
1,060,99
1,261,17
0,980,99
0,771,16
0,750,66
mo
del11
0,71,24
1,211,01
1,020,89
0,520,38
0,560,54
mo
del12
0,71,22
1,170,99
1,001,18
0,870,73
0,971,02
n ∗1535
15101485
14601435
14101385
13601335
1310f
2550
75100
125150
175200
225250
mo
del1
1,001,00
1,001,00
1,001,00
1,001,00
1,001,00
mo
del2
1,041,05
0,911,00
0,980,91
0,90,99
0,940,97
mo
del3
1,141,09
0,70,76
1,281,10
1,231,04
1,421,85
mo
del4
1,121,17
0,780,85
1,361,21
1,371,12
1,592,03
mo
del5
1,121,05
0,310,31
0,670,5
0,480,46
0,670,96
mo
del6
0,870,68
0,310,31
0,590,52
0,490,46
0,671,02
mo
del7
1,190,6
0,380,36
0,570,51
0,680,87
1,031,29
mo
del8
0,910,51
0,380,35
0,540,48
0,620,78
0,971,17
mo
del9
1,121,05
0,310,31
0,670,5
0,480,46
0,670,96
mo
del10
1,111,09
0,40,41
0,90,8
0,710,66
0,971,89
mo
del11
0,851,14
0,590,66
1,231,21
1,631,43
1,421,80
mo
del12
1,101,08
0,710,75
1,241,09
1,481,53
1,361,91
Ratio
sb
etween
R1f
(perfo
rman
cem
easure
(3.35)fo
rm
od
el1)an
dR
jf(p
erform
ance
measu
re(3.35)
for
mo
del
j),j=
1,...,12fo
rd
ifferentvalu
eso
fn ∗(th
elen
gtho
fthe
estimatio
nsam
ple)
and
f(th
en
um
ber
ofo
ut-o
f-samp
lefo
recasts).SeeSectio
n3.4
for
mo
red
etailso
nth
efo
recasting
pro
cedu
re.Th
eco
mp
leted
atasetis
mad
eu
po
fn=
n ∗+f=
1560o
bservatio
ns,ran
ging
from
Janu
ary1880
toD
ecemb
er2009.
3.9. APPENDIX 115
Tab
le3.
2.R
atio
sb
etw
een
R2 f
and
Rj f
,j=
1,..
.,12
.
n∗
1535
1510
1485
1460
1435
1410
1385
1360
1335
1310
f25
5075
100
125
150
175
200
225
250
mo
del
11,
000,
980,
981,
000,
990,
961,
120,
940,
980,
98m
od
el2
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
mo
del
30,
991,
081,
051,
031,
000,
981,
140,
941,
001,
01m
od
el4
0,99
1,07
1,04
1,02
0,98
0,95
1,14
0,84
0,95
1,01
mo
del
51,
070,
981,
281,
201,
171,
191,
190,
711,
071,
02m
od
el6
1,06
0,96
1,23
1,17
1,06
0,78
0,7
0,95
0,54
0,49
mo
del
70,
71,
231,
211,
091,
151,
090,
850,
520,
990,
95m
od
el8
0,7
1,23
1,20
1,08
1,14
1,17
1,34
1,08
1,00
0,89
mo
del
91,
070,
981,
281,
201,
171,
191,
190,
711,
071,
02m
od
el10
1,06
0,97
1,24
1,16
0,97
0,95
0,87
1,09
0,73
0,65
mo
del
110,
71,
221,
181,
011,
000,
850,
580,
360,
550,
53m
od
el12
0,7
1,19
1,14
0,99
0,99
1,13
0,98
0,69
0,94
1,00
n∗
1535
1510
1485
1460
1435
1410
1385
1360
1335
1310
f25
5075
100
125
150
175
200
225
250
mo
del
10,
960,
951,
101,
001,
021,
101,
111,
011,
061,
03m
od
el2
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
mo
del
31,
101,
040,
770,
761,
301,
211,
371,
051,
511,
91m
od
el4
1,07
1,12
0,86
0,85
1,39
1,33
1,52
1,13
1,69
2,09
mo
del
51,
071,
000,
340,
310,
680,
550,
530,
460,
710,
98m
od
el6
0,83
0,65
0,34
0,31
0,6
0,58
0,55
0,46
0,71
1,05
mo
del
71,
140,
570,
420,
360,
580,
570,
760,
871,
101,
33m
od
el8
0,87
0,49
0,42
0,35
0,55
0,53
0,69
0,79
1,03
1,20
mo
del
91,
071,
000,
340,
310,
680,
550,
540,
460,
710,
99m
od
el10
1,07
1,03
0,44
0,41
0,92
0,88
0,79
0,67
1,03
1,95
mo
del
110,
821,
080,
640,
661,
251,
331,
811,
441,
511,
86m
od
el12
1,06
1,03
0,78
0,75
1,26
1,20
1,65
1,54
1,44
1,97
Rat
ios
bet
wee
nR
2 f(p
erfo
rman
cem
easu
re(3
.35)
for
mo
del
2)an
dR
j f(p
erfo
rman
cem
easu
re(3
.35)
for
mo
del
j),
j=
1,..
.,12
for
dif
fere
ntv
alu
eso
fn∗
(th
ele
ngt
ho
fth
e
esti
mat
ion
sam
ple
)an
df
(th
en
um
ber
ofo
ut-
of-
sam
ple
fore
cast
s).S
eeSe
ctio
n3.
4fo
rm
ore
det
ails
on
the
fore
cast
ing
pro
ced
ure
.Th
eco
mp
lete
dat
aset
ism
ade
up
of
n=
n∗ +
f=
1560
ob
serv
atio
ns,
ran
gin
gfr
om
Jan
uar
y18
80to
Dec
emb
er20
09.
116 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Table
3.3.Ratio
sb
etween
R3f
and
Rjf ,
j=1,...,12.
n ∗1535
15101485
14601435
14101385
13601335
1310f
2550
75100
125150
175200
225250
mo
del1
1,010,91
0,930,97
0,990,97
0,981,00
0,980,97
mo
del2
1,010,93
0,950,97
1,001,02
0,881,06
1,000,99
mo
del3
1,001,00
1,001,00
1,001,00
1,001,00
1,001,00
mo
del4
1,001,00
0,990,99
0,980,96
1,000,9
0,951,00
mo
del5
1,070,91
1,221,17
1,171,21
1,040,75
1,071,01
mo
del6
1,070,9
1,171,13
1,060,8
0,611,01
0,540,48
mo
del7
0,711,15
1,151,06
1,151,11
0,750,55
0,990,95
mo
del8
0,711,14
1,141,04
1,141,19
1,171,15
1,010,88
mo
del9
1,070,91
1,221,17
1,171,21
1,040,75
1,071,01
mo
del10
1,070,9
1,181,13
0,970,96
0,761,16
0,730,65
mo
del11
0,711,13
1,130,98
1,000,86
0,510,38
0,550,52
mo
del12
0,711,11
1,090,96
0,981,15
0,860,73
0,950,99
n ∗1285
12601235
12101185
11601135
11101085
1060f
275300
325350
375400
425450
475500
mo
del1
0,870,91
1,421,31
0,780,91
0,810,96
0,70,54
mo
del2
0,910,96
1,301,31
0,770,83
0,730,95
0,660,52
mo
del3
1,001,00
1,001,00
1,001,00
1,001,00
1,001,00
mo
del4
0,981,07
1,111,12
1,071,10
1,111,07
1,121,10
mo
del5
0,980,96
0,440,4
0,520,45
0,390,44
0,470,52
mo
del6
0,760,62
0,450,4
0,460,47
0,40,44
0,470,55
mo
del7
1,040,54
0,540,48
0,450,47
0,560,83
0,730,7
mo
del8
0,80,47
0,540,46
0,420,44
0,50,75
0,680,63
mo
del9
0,980,96
0,440,4
0,520,45
0,390,44
0,470,52
mo
del10
0,970,99
0,570,54
0,710,72
0,580,63
0,681,02
mo
del11
0,751,04
0,840,86
0,961,10
1,321,37
1,000,97
mo
del12
0,970,99
1,010,98
0,970,99
1,201,46
0,961,03
Ratio
sb
etween
R3f
(perfo
rman
cem
easure
(3.35)fo
rm
od
el3)an
dR
jf(p
erform
ance
measu
re(3.35)
for
mo
del
j),j=
1,...,12fo
rd
ifferentvalu
eso
fn ∗(th
elen
gtho
fthe
estimatio
nsam
ple)
and
f(th
en
um
ber
ofo
ut-o
f-samp
lefo
recasts).SeeSectio
n3.4
for
mo
red
etailso
nth
efo
recasting
pro
cedu
re.Th
eco
mp
leted
atasetis
mad
eu
po
fn=
n ∗+f=
1560o
bservatio
ns,ran
ging
from
Janu
ary1880
toD
ecemb
er2009.
3.9. APPENDIX 117
Tab
le3.
4.R
atio
sb
etw
een
R4 f
and
Rj f
,j=
1,..
.,12
.
n∗
1535
1510
1485
1460
1435
1410
1385
1360
1335
1310
f25
5075
100
125
150
175
200
225
250
mo
del
11,
010,
920,
940,
981,
011,
010,
981,
121,
030,
97m
od
el2
1,01
0,93
0,96
0,98
1,02
1,06
0,88
1,18
1,06
0,99
mo
del
31,
001,
001,
011,
011,
031,
041,
001,
121,
061,
00m
od
el4
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
1,00
mo
del
51,
070,
911,
231,
181,
201,
261,
040,
841,
131,
01m
od
el6
1,07
0,9
1,18
1,14
1,08
0,83
0,61
1,12
0,57
0,48
mo
del
70,
711,
151,
161,
071,
181,
150,
750,
611,
050,
95m
od
el8
0,71
1,15
1,15
1,05
1,16
1,24
1,17
1,28
1,06
0,88
mo
del
91,
070,
911,
231,
181,
201,
261,
040,
841,
131,
01m
od
el10
1,07
0,9
1,19
1,14
0,99
1,00
0,76
1,30
0,77
0,64
mo
del
110,
711,
141,
140,
991,
030,
90,
510,
420,
580,
52m
od
el12
0,71
1,11
1,09
0,97
1,01
1,19
0,86
0,82
1,00
0,99
n∗
1535
1510
1485
1460
1435
1410
1385
1360
1335
1310
f25
5075
100
125
150
175
200
225
250
mo
del
10,
890,
851,
281,
170,
730,
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118 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Table 3.5. Parameter estimates for model 1-2.
(a) Continuous-time, model 1-2, parameter estimates.
σT aSS aST aT S aT T[cK
] [1
month
] [mm
cK ·month
] [cK
mm·month
] [1
month
]value 7,59 -0,0112 0,0056 0,0512 -0,0816
std. (0,2311) (0,0016) (0,002) (0,0075) (0,0075)t-ratio [32,84] [-7] [2,8] [6,83] [-10,88]
λSl
λTl
√ΣSS ΣST
√ΣT T[
mmmonth
] [cK
month
] [mmp
month
] [mm·cKmonth
] [cKp
month
]value 0,0012 -0,0022 1,11 0,34 5,87
std. (0,0001) (0,0008) (0,1151) (0,3733) (0,2703)t-ratio [12] [-2,75] [9,64] [0,91] [21,72]
Parameter estimates for model 1-2 in the continuous-time representation (equations (3.22), (3.24), and(3.32)). Standard deviations are reported in parentheses, and the t-ratios in square brackets. The parameterswere estimated using information from time t = 1 (January 1880) to time t = n (December 2009). Thestandard deviations were obtained from the bootstrap procedure described in Appendix 3.9.
(b) Continuous-time, model 1-2, parameter estimates (alternative measurement units).
σT aSS aST aT S aT T[K
] [1
year
] [mm
cK ·month
] [cK
mm·month
] [1
year
]value 0,0759 -0,1344 0,00672 6,144 -0,9792
λSl
λTl
√ΣSS ΣST
√ΣT T[
myear
] [K
year
] [mpyear
] [m·Kyear
] [Kp
year
]value 0,000014 -0,00026 0,0038 0,000041 0,20
Parameter estimates for model 1-2 in the continuous-time representation (equations (3.22), (3.24), and(3.32)). The parameters were estimated using information from time t = 1 (January 1880) to time t = n(December 2009).
3.9. APPENDIX 119
Table 3.6. Parameter estimates for model 1-2.
(a) Discrete-time, model 1-2, parameter estimates.
σT A∗,SS A∗,ST A∗,SµSA∗,SµT
A∗,T S A∗,T T A∗,TµS[cK
] [mmcK
] [mmcK
] [cK
mm
] [cK
mm
]value 7,59 0,99 0,0054 0,99 0,0027 0,0489 0,92 0,0248
std. (0,2311) (0,0015) (0,0019) (0,0008) (0,001) (0,007) (0,0068) (0,0036)t-ratio [32,82] [641,96] [2,86] [1275,33] [2,86] [6,98] [135,24] [6,92]
A∗,TµTcS cT cµ
Scµ
T √Σ∗,SS Σ∗,ST
√Σ∗,T T
[mm][cK
][mm]
[cK
][mm]
[mm · cK
] [cK
]value 0,96 0,0006 -0,001 0,0012 -0,0022 1,11 0,44 5,64
std. (0,0035) (0,0001) (0,0004) (0,0001) (0,0008) (0,1108) (0,3661) (0,2535)t-ratio [273,14] [8,28] [-2,56] [8,25] [-2,57] [9,98] [1,21] [22,24]
Parameter estimates for model 1-2 in the dicrete-time representation (equations (3.34)). Standard devi-ations are reported in parentheses, and the t-ratios in square brackets. The parameters were estimatedusing information from time t = 1 (January 1880) to time t = n (December 2009). The standard deviationswere obtained from the bootstrap procedure described in Appendix 3.9.
(b) Discrete-time, model 1-2, parameter estimates (alternative measurement units).
σT A∗,SS A∗,ST A∗,SµSA∗,SµT
A∗,T S A∗,T T A∗,TµS[K
] [mK
] [mK
] [Km
] [Km
]value 0,0759 0,99 0,00054 0,99 0,00027 0,489 0,92 0,248
A∗,TµTcS cT cµ
Scµ
T √Σ∗,SS Σ∗,ST
√Σ∗,T T
[m][K
][m]
[K
][m]
[m ·K
] [K
]value 0,96 0,0000006 -0,00001 0,0000012 -0,000022 0,00111 0,0000044 0,0564
Parameter estimates for model 1-2 in the discrete-time representation (equations (3.34)). The parameterswere estimated using information from time t = 1 (January 1880) to time t = n (December 2009).
120 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Table 3.7. Parameter estimates for model 3-4.
(a) Continuous-time, model 3-4, parameter estimates.
σT aSS aST aT S aT T[cK
] [1
month
] [mm
cK ·month
] [cK
mm·month
] [1
month
]value 7,41 -0,0288 -0,0037 -0,0458 -0,1169
std. (0,27) (0,0014) (0,0026) (0,006) (0,0106)t-ratio [27,44] [-20,57] [-1,42] [-7,63] [-11,03]
λSq λT
q
√ΣSS ΣST
√ΣT T[
mmmonth3
] [cK
month3
] [mmp
month
] [mm·cKmonth
] [cKp
month
]value 0,000003 0,000015 1,25 1,46 5,91
std. (0,0000004) (0,000002) (0,19) (0,47) (0,30)t-ratio [7,5] [7,5] [6,43] [3,12] [19,97]
Parameter estimates for model 3-4 in the continuous-time representation (equations (3.22), (3.25), and(3.32)). Standard deviations are reported in parentheses, and the t-ratios in square brackets. The parameterswere estimated using information from time t = 1 (January 1880) to time t = n (December 2009). Thestandard deviations were obtained from the bootstrap procedure described in Appendix 3.9.
(b) Continuous-time, model 3-4, parameter estimates (alternative measurement units).
σT aSS aST aT S aT T[K
] [1
year
] [m
K ·year
] [K
m·year
] [1
year
]value 0,0741 -0,3456 -0,00444 -5,496 -1,4028
λSq λT
q
√ΣSS ΣST
√ΣT T[
myear 3
] [K
year 3
] [mpyear
] [m·Kyear
] [Kp
year
]value 0,00000043 0,000021 0,0043 0,00018 0,20
Parameter estimates for model 3-4 in the continuous-time representation (equations (3.22), (3.25), and(3.32)). The parameters were estimated using information from time t = 1 (January 1880) to time t = n(December 2009).
3.9. APPENDIX 121
Table 3.8. Discrete-time, model 3-4, parameter estimates.
σT A∗,SS A∗,ST A∗,SµSA∗,SµT
A∗,SλS[cK
] [mmcK
] [mmcK
] [month
]value 7,41 0,97 -0,0035 0,99 -0,0018 0,4952
std. (0,27) (0,0013) (0,0024) (0,0007) (0,0012) (0,0002)t-ratio [27,44] [746,15] [-1,46] [1414,29] [-1,5] [2476]
A∗,SλTA∗,T S A∗,T T A∗,TµS
A∗,TµTA∗,TλS[
month·mmcK
] [cK
mm
] [cK
mm
] [month·cK
mm
]value -0,0006 -0,0426 0,89 -0,0218 0,94 -0,0074
std. (0,0004) (0,0053) (0,0092) (0,0028) (0,0048) (0,0009)t-ratio [-1,5] [-8,04] [96,74] [-7,79] [196,63] [-8,22]
A∗,TλTcS cT cµ
Scµ
T[month
][mm]
[cK
][mm]
[cK
]value 0,4811 0,00000044 0,000002 0,000001 0,000007
std. (0,0016) (0,00000007) (0,00000039) (0,0000002) (0,000001)t-ratio [300,69] [6,29] [6,05] [6,65] [7,32]
cλS
cλT √
Σ∗,SS Σ∗,ST√Σ∗,T T[
mmmonth
] [cK
month
][mm]
[mm · cK
] [cK
]value 0,000003 0,000015 1,25 1,46 5,91
std. (0,0000004) (0,000002) (0,19) (0,47) (0,3)t-ratio [6,63] [7,32] [6,58] [3,11] [19,7]
Parameter estimates for model 3-4 in the discrete-time representation (equations (3.22), (3.25), and (3.32)).Standard deviations are reported in parentheses, and the t-ratios in square brackets. The parameters wereestimated using information from time t = 1 (January 1880) to time t = n (December 2009). The standarddeviations were obtained from the bootstrap procedure described in Appendix 3.9.
122 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL
Table 3.9. Discrete-time, model 3-4, parameter estimates (alternative measurement units).
σT A∗,SS A∗,ST A∗,SµSA∗,SµT
A∗,SλS[K
] [mK
] [mK
] [year
]value 0,074 0,97 -0,00035 0,99 -0,00018 0,041
A∗,SλTA∗,T S A∗,T T A∗,TµS
A∗,TµTA∗,TλS[
year ·mK
] [Km
] [Km
] [year ·K
m
]value -0,000005 -0,426 0,89 -0,218 0,94 -0,0062
A∗,TλTcS cT cµ
Scµ
T[year
][m]
[K
][m]
[K
]value 0,040 0,00000000044 0,00000002 0,000000001 0,00000007
cλS
cλT √
Σ∗,SS Σ∗,ST√Σ∗,T T[
myear
] [K
year
][m]
[m ·K
] [K
]value 0,000000036 0,0000018 0,0013 0,000014 0,0591
Parameter estimates for model 3-4 in the discrete-time representation (equations (3.22), (3.25), and (3.32)).The parameters were estimated using information from time t = 1 (January 1880) to time t = n (December2009).
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